Understanding Quadratic Functions

1. What is a Quadratic Function?

What Is a Quadratic Function?

A quadratic function is a polynomial function in which the highest power of the variable is 2.

A quadratic function is usually written in standard form:

f(x) =ax² + bx + c​

where:

• a, b, and c are constants.
• a ≠ 0.
• x is the variable.
• ax² is the quadratic term.
• bx is the linear term.
• c is the constant term.

The condition a ≠ 0 is important. If a = 0, the x² term disappears and the function is no longer quadratic.

Example

Consider:

f(x) = 2x² + 5x − 3

In this function:

a = 2, b = 5, c = −3

Because the highest power of x is 2, this is a quadratic function.

Recognising Standard Form

Quadratic functions do not need to contain all three terms.

For example, all of these are quadratic:

y = x² + 4x + 7

y = 3x² − 5

y = −2x² + 6x

y = 5x²

Each function is quadratic because its highest power of x is 2.

The terms may also appear in a different order. For example:

y = 7 + 3x − 2x²

can be rearranged into standard form:

y = −2x² + 3x + 7

This makes it easier to identify:

a = −2, b = 3, c = 7

The Degree of a Polynomial

The degree of a polynomial is the highest exponent of its variable.

For example:

A quadratic function therefore has degree 2.

Quick Check

What is the degree of:

f(x) = 7x² − 3x + 9

The highest exponent is 2.

Therefore:

Degree = 2​

The function is quadratic.

Quadratic or Not?

One of the easiest ways to identify a quadratic function is to look for its highest exponent.

Consider:

y = 4x + 1

The highest exponent is 1, so this is linear, not quadratic.

Now consider:

y = 4x² + 1

The highest exponent is 2, so this is quadratic.

Finally:

y = 4x³ + x2 + 1

Although this equation contains an x2 term, its highest exponent is 3. It is therefore a cubic function, not a quadratic function.

Important

Simply seeing x² does not automatically mean that a function is quadratic.

The x² term must contain the highest power of x.

Linear Functions vs Quadratic Functions

A linear function has the general form:

y = mx + b

Its graph is a straight line.

A quadratic function has the general form:

y = ax² + bx + c

Its graph is a curve called a parabola.

For example:

Linear:

y = 2x + 1

Quadratic:

y = x²

For y = x², a few values are:

Notice that the y-values do not increase at a constant rate. This changing rate produces a curved graph rather than a straight line.

The Parabola

The graph of every quadratic function is a parabola.

A simple example is:

y = x²

Its graph is U-shaped and has a lowest point at (0, 0).

For a general quadratic,

y = ax² + bx + c

the value of a helps determine the direction of the parabola.

• If a > 0, the parabola opens upward.
• If a < 0, the parabola opens downward.

For example:

y = x²

opens upward, while

y = −x²

opens downward.

The turning point of a parabola is called its vertex.

Why Do Quadratic Functions Form Parabolas?

In a linear function, equal changes in x produce a constant change in y. This produces a straight line.

In a quadratic function, the x² term causes the rate of change to change as x changes.

For y = x²:

0, 1, 4, 9, 16, 25,…

The first differences are:

1, 3, 5, 7, 9,…

These differences are changing.

However, the differences between those differences are constant:

2, 2, 2, 2,…

This pattern of constant second differences is a characteristic of quadratic functions and corresponds to the curved shape of a parabola.

Quadratics in the Real World

Quadratic functions are useful because many situations involve quantities that change at a changing rate.

For example, the approximate path of a thrown ball can be modelled by a quadratic function.

 

 

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Quadratic models can appear when studying:

• The path of a thrown or launched object.
• The height of an object over time.
• The shape of some arches and reflectors.
• Maximum-area and minimum-cost problems.
• Relationships involving acceleration.

Worked Example

Identify whether each function is quadratic.

A.

f(x) = 3x² − 7x + 2

The highest exponent is 2.

Quadratic​

B.

f(x) = 5x − 8

The highest exponent is 1.

Linear​

C.

f(x) = x³ + 2x2 − 4

The highest exponent is 3.

Cubic​

D.

f(x) = −6x² + 9

The highest exponent is 2.

Quadratic​

Key Vocabulary

Quadratic function – A polynomial function with degree 2.

Standard form – The form f(x) = ax² + bx + c where a ≠ 0.

Degree – The highest exponent of the variable in a polynomial.

Polynomial – An expression made from variables, constants, and non-negative whole-number exponents.

Parabola – The curved graph produced by a quadratic function.

Vertex – The turning point of a parabola.

Coefficient – A number multiplying a variable.

Key Takeaways

• A quadratic function has degree 2.
• Its standard form is: f(x) = ax² + bx + c,a ≠ 0​
• The highest power of x must be 2.
• A linear function has degree 1, while a quadratic function has degree 2.
• Having an x² term does not make a function quadratic if a higher-power term is also present.
• The graph of a quadratic function is called a parabola.
• Parabolas open upward when a > 0 and downward when a < 0.
• Quadratic functions have constant second differences when x-values are equally spaced.

2. Features of a Parabola

Learning outcomes

• I can identify the vertex of a parabola.
• I can identify the axis of symmetry.
• I can determine whether a parabola opens upward or downward.
• I can identify maximum and minimum points.
• I can describe the overall shape of a quadratic graph.

What Is a Parabola?

The graph of a quadratic function is called a parabola.

A quadratic function can be written in standard form:

y = ax² + bx + c

A parabola is a smooth, symmetrical curve. Depending on the quadratic function, it may open upward or downward.

 

 

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Several important features can be identified from a parabola:

• Vertex
• Axis of symmetry
• Direction of opening
• Maximum or minimum point
• Intercepts

In this lesson, we will focus on the first four.

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The Vertex

The vertex is the turning point of a parabola.

It is the point where the graph changes direction.

For example, consider a parabola with its vertex at:

(2, −3)

The point (2, −3) is the vertex.

The vertex is especially important because it represents either the minimum or maximum value of the quadratic function.

Remember

The vertex is written as a coordinate:

(x, y)​

So if the vertex is (−1, 4):

• x = −1
• y = 4

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The Axis of Symmetry

Every parabola is symmetrical.

This means that one side of the parabola is a mirror image of the other.

The vertical line passing through the vertex is called the axis of symmetry.

If the vertex is:

(3, −2)

then the axis of symmetry is:

x = 3​

Notice that the axis of symmetry uses the x-coordinate of the vertex.

Example

Vertex:

(−4, 5)

Axis of symmetry:

x = −4​

The axis of symmetry divides the parabola into two equal halves.

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Upward-Opening Parabolas

Some parabolas open upward, producing a shape similar to the letter U.

For example:

y = x²

opens upward.

In the standard form

y = ax² + bx + c

look at the value of a.

If:

a > 0​

the parabola opens upward.

Examples include:

y = 2x² + 3x − 1 

y = x² − 6 

y = 5x² − 4x + 2

In each case, the coefficient of x² is positive.

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Downward-Opening Parabolas

A parabola can also open downward.

For example:

y = −x²

opens downward.

If:

a < 0​

the parabola opens downward.

Examples include:

y = −2x² + 5x + 3

y = −x² + 4

y = −7x² + 2x − 1

In each case, the coefficient of x² is negative.

Quick Rule

Value of a	Direction
a > 0	Opens upward
a < 0	Opens downward

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Minimum Points

When a parabola opens upward, the vertex is the lowest point on the graph.

This is called the minimum point.

For example, suppose the vertex is:

(2, −4)

and the parabola opens upward.

The minimum point is:

(2, −4)​

The minimum value of the function is therefore:

y = −4​

Be careful with the language:

• Minimum point: (2, −4)
• Minimum value: −4

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Maximum Points

When a parabola opens downward, the vertex is the highest point on the graph.

This is called the maximum point.

Suppose the vertex is:

(−1, 6)

and the parabola opens downward.

The maximum point is:

(−1, 6)​

The maximum value is:

y = 6​

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Connecting the Features

The vertex, axis of symmetry, direction of opening, and maximum or minimum are all connected.

Suppose a parabola has:

Vertex = (3, −5)

and opens upward.

We immediately know:

Vertex

(3, −5)

Axis of symmetry

x = 3

Direction

Upward

Type of turning point

Minimum

Minimum value

y = −5

A lot of information about a quadratic graph can therefore be found from just its vertex and direction.

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Describing the Shape of a Parabola

When describing a quadratic graph, you can mention several features.

For example:

The graph is a parabola that opens upward. Its vertex is at (2, −3), so it has a minimum point at (2, −3). Its axis of symmetry is x = 2.

This gives a clear mathematical description of the graph.

A useful checklist is:

1. Does it open upward or downward?
2. Where is the vertex?
3. What is the axis of symmetry?
4. Does the graph have a maximum or minimum?

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Worked Example 1

Consider:

y = (x − 2)² − 3

The graph has a vertex at:

(2, −3)​

The coefficient of the squared expression is positive, so the parabola opens upward.

Therefore:

• Vertex: (2, −3)
• Axis of symmetry: x = 2
• Direction: upward
• Turning point: minimum
• Minimum value: −3

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Worked Example 2

Consider:

y = −(x + 1)² + 4

The vertex is:

(−1, 4)​

The negative sign tells us that the parabola opens downward.

Therefore:

• Vertex: (−1, 4)
• Axis of symmetry: x = −1
• Direction: downward
• Turning point: maximum
• Maximum value: 4

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The Width of a Parabola

The value of a also affects how wide or narrow a parabola appears.

Compare:

y=x²

and

y = 3x²

The graph of y = 3x² is narrower.

Now compare:

y = x²

and

y = \( \frac{1}{2}x^2 \)

The graph of y = \( \frac{1}{2}x^2 \) is wider.

In general:

• Larger ∣a∣ → narrower parabola
• Smaller ∣a∣ → wider parabola

The sign of a controls the direction, while the size of ∣a∣ affects the width.

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Parabolas in Real-World Situations

Parabolic shapes and quadratic models appear in many situations.

 

 

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For example, the height of a ball thrown into the air can often be modelled using a quadratic function.

The ball rises, reaches a maximum height, and then falls.

The maximum point of the graph represents the greatest height reached by the ball.

This is one reason why identifying the vertex of a parabola is useful.

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Key Vocabulary

Parabola – The curved graph of a quadratic function.

Vertex – The turning point of a parabola.

Axis of symmetry – The vertical line that divides a parabola into two mirror-image halves.

Maximum point – The highest point of a downward-opening parabola.

Minimum point – The lowest point of an upward-opening parabola.

Maximum value – The y-coordinate of a maximum point.

Minimum value – The y-coordinate of a minimum point.

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Key Takeaways

• The graph of a quadratic function is called a parabola.
• The vertex is the turning point of the parabola.
• The axis of symmetry is the vertical line passing through the vertex.
• If the vertex is (h, k), the axis of symmetry is: x = h​
• If a > 0, the parabola opens upward and has a minimum.
• If a < 0, the parabola opens downward and has a maximum.
• The vertex gives the location of the maximum or minimum point.
• The two sides of a parabola are symmetrical.
• The value of ∣a∣ affects how wide or narrow the parabola appears.

 

 

 

3. Intercepts

What are intercepts?

Intercepts are points where a graph meets the coordinate axes. They help us connect an equation to its graph and understand what the function represents.

A quadratic function can be written in the form:

y = ax² + bx + c, where a ≠ 0

Its graph is a curved shape called a parabola.

• If a > 0, the parabola opens upwards and has a minimum point.
• If a < 0, the parabola opens downwards and has a maximum point.

A quadratic graph can meet the x-axis twice, once, or not at all. However, a quadratic function with an unrestricted domain always has exactly one y-intercept.

Identifying intercepts from a graph

The x-intercepts

An x-intercept is a point where the graph meets the x-axis.

Every point on the x-axis has a y-coordinate of zero. Therefore, an x-intercept has the form:

(x, 0)

For example, if a parabola crosses the x-axis at x = −1 and x = 3, its x-intercepts are:

(−1, 0) and (3, 0)

The corresponding x-values are called the zeros of the function or the roots of the equation f(x) = 0.

The y-intercept

The y-intercept is the point where the graph crosses the y-axis.

Every point on the y-axis has an x-coordinate of zero. Therefore, a y-intercept has the form:

(0, y)

If a parabola crosses the y-axis at y = −3, its y-intercept is:

(0, −3)

When reading a graph, check the scale carefully. One grid square does not always represent one unit.

 

 

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The orange points show x-intercepts. The pink points show y-intercepts. Notice that the middle graph touches the x-axis without crossing it: this still counts as an intercept.

Finding the y-intercept from an equation

To find a y-intercept, substitute x = 0.

For a quadratic in the form y = ax² + bx + c:

y = a(0)² + b(0) + c

y = c

Therefore, the y-intercept is (0, c).

This shortcut applies when the equation is written in expanded form.

Worked example: finding a y-intercept

Find the y-intercept of:

y = 2x² − 5x + 3

Substitute x = 0:

y = 2(0)² − 5(0) + 3
y = 3

The y-intercept is (0, 3).

Worked example: an equation in factored form

Find the y-intercept of:

y = (x − 4)(x + 2)

Substitute x = 0:

y = (0 − 4)(0 + 2)
y = (−4)(2)
y = −8

The y-intercept is (0, −8).

There is no need to expand the brackets first.

Finding x-intercepts by factorization

To find x-intercepts, substitute y = 0 and solve for x.

If the quadratic can be factorized, use the zero-product property:

If two factors multiply to give zero, at least one factor must equal zero.

For example, if:

(x − 3)(x + 1) = 0

then:

x − 3 = 0 or x + 1 = 0

Therefore:

x = 3 or x = −1

Worked example: finding both types of intercept

Find the intercepts of:

y = x² − 2x − 3

Find the y-intercept.

Substitute x = 0:

y = 0² − 2(0) − 3
y = −3

The y-intercept is (0, −3).

Find the x-intercepts.

Substitute y = 0:

x² − 2x − 3 = 0

Find two numbers that multiply to −3 and add to −2. These are −3 and +1.

Factorize:

(x − 3)(x + 1) = 0

Set each factor equal to zero:

x − 3 = 0 or x + 1 = 0

x = 3 or x = −1

The x-intercepts are (3, 0) and (−1, 0).

These are the intercepts shown on the left-hand graph above.

Finding x-intercepts using square roots

Some equations can be solved directly by isolating a squared expression.

Worked example

Find the x-intercepts of:

y = (x − 2)² − 9

Set y = 0:

0 = (x − 2)² − 9

Add 9 to both sides:

(x − 2)² = 9

Take both the positive and negative square roots:

x − 2 = ±3

Therefore:

x = 2 + 3 or x = 2 − 3

x = 5 or x = −1

The x-intercepts are (5, 0) and (−1, 0).

Remember both square roots. Both 3² and (−3)² equal 9. Using only the positive square root would miss one intercept.

Finding x-intercepts using the quadratic formula

Not every quadratic factorizes easily. The quadratic formula can solve any equation of the form:

ax² + bx + c = 0

The formula is:

x = [−b ± √(b² − 4ac)] / (2a)

Worked example

Find the x-intercepts of:

y = x² − 2x − 1

Set y = 0:

x² − 2x − 1 = 0

Identify the coefficients:

a = 1, b = −2, c = −1

Substitute:

x = [2 ± √(4 + 4)] / 2
x = [2 ± √8] / 2
x = 1 ± √2

The exact x-intercepts are:

(1 − √2, 0) and (1 + √2, 0)

Approximately, these are:

(−0.414, 0) and (2.414, 0)

A graph may give approximate intercepts, while algebra can give exact values.

Why can there be two, one, or no x-intercepts?

Finding x-intercepts means finding real solutions to:

ax² + bx + c = 0

A quadratic equation can have two distinct real solutions, one repeated real solution, or no real solutions.

Two x-intercepts

The parabola crosses the x-axis at two different points.

For example:

y = x² − 2x − 3

Setting y = 0 gives:

(x − 3)(x + 1) = 0

There are two distinct roots: x = 3 and x = −1.

One x-intercept

The vertex lies on the x-axis, so the parabola touches the axis at one point.

For example:

y = (x − 1)²

Setting y = 0 gives:

(x − 1)² = 0

x = 1

The only x-intercept is (1, 0).

This is called a repeated root because the factor (x − 1) appears twice.

No x-intercepts

The parabola lies entirely above or entirely below the x-axis.

For example:

y = (x − 1)² + 2

A squared real number cannot be negative. Therefore:

(x − 1)² ≥ 0

Adding 2 means y is always at least 2. The graph never reaches y = 0, so it has no x-intercepts.

It still has a y-intercept: (0, 3).

Using the discriminant

The expression inside the square root in the quadratic formula is called the discriminant:

D = b² − 4ac

It tells us how many real x-intercepts a quadratic has without requiring us to solve the equation fully.

Worked example

How many x-intercepts does this function have?

y = 2x² + 4x + 5

Here, a = 2, b = 4 and c = 5.

D = 4² − 4(2)(5)
D = 16 − 40
D = −24

Because the discriminant is negative, the graph has no x-intercepts.

“No real roots” means there are no solutions on the real number line. It does not mean that the function has no graph.

Sketching a parabola using its intercepts

Intercepts provide useful points for a sketch. To show the shape accurately, also identify the opening direction, axis of symmetry and vertex.

Worked example: sketch y = x² − 2x − 3

Find the intercepts.

From the earlier calculation:

• x-intercepts: (−1, 0) and (3, 0).
• y-intercept: (0, −3).

Determine the opening direction.

The coefficient of x² is positive, so the parabola opens upwards.

Find the axis of symmetry.

When there are two x-intercepts, the axis of symmetry lies halfway between them:

x = (−1 + 3) / 2
x = 1

Find the vertex.

Substitute x = 1 into the equation:

y = 1² − 2(1) − 3
y = −4

The vertex is (1, −4).

Use symmetry to find another point.

The point (0, −3) is one unit to the left of x = 1. Its reflection is one unit to the right:

(2, −3)

Draw the curve.

Plot the points and draw a smooth, symmetrical curve through them. The curve should turn at (1, −4) and continue upwards on both sides.

Do not join the points with straight line segments. A parabola is a smooth curve.

What if there are no x-intercepts?

Use the y-intercept, vertex and additional symmetrical points.

For y = (x − 1)² + 2:

• Vertex: (1, 2).
• Axis of symmetry: x = 1.
• y-intercept: (0, 3).
• Reflected point: (2, 3).
• Opening direction: upwards.

These features are enough for a useful sketch, even though there are no x-intercepts.

What do intercepts represent in real situations?

The meaning of an intercept depends on what the axes measure.

The y-intercept represents the output when the input is zero.

For example, it might represent:

• The initial height of a thrown object.
• A company's profit or loss when no items are sold.
• The value of a measurement at the start of an investigation.

An x-intercept represents an input for which the output is zero.

For example, it might represent:

• The time when an object reaches ground level.
• A break-even quantity where profit is zero.
• A position where a modelled height is zero.

Worked example: the height of a ball

A simplified model gives a ball’s height as:

h = −5t² + 20t

Here, h is height in metres and t is time in seconds.

Find the vertical intercept.

At t = 0:

h = 0

The intercept (0, 0) means the ball starts at ground level.

Find the horizontal intercepts.

Set h = 0:

−5t² + 20t = 0
−5t(t − 4) = 0

Therefore:

t = 0 or t = 4

The intercepts mean the ball is at ground level at launch and returns to ground level after 4 seconds.

For this flight, the meaningful domain is 0 ≤ t ≤ 4. The equation can be graphed outside this interval, but those parts do not describe the ball’s flight.

Common misconceptions

• “To find an x-intercept, set x = 0.” Set y = 0. Setting x = 0 finds the y-intercept.
• “Every parabola has two x-intercepts.” A parabola can have two, one, or no x-intercepts.
• “Touching the x-axis does not count.” A point of contact is an x-intercept, even if the curve does not cross the axis.
• “The factor (x + 3) gives the root x = 3.” Solving x + 3 = 0 gives x = −3.
• “The vertex is always the y-intercept.” This happens only when the axis of symmetry is the y-axis.
• “Every algebraic intercept is meaningful in a model.” Context may exclude values such as negative times.

Did you know?

For a quadratic with two distinct x-intercepts, the vertex is always horizontally halfway between them.

If the roots are r₁ and r₂, the axis of symmetry is:

x = (r₁ + r₂) / 2

This gives a quick way to locate the vertex’s x-coordinate. You can then substitute that value into the equation to find its y-coordinate.

Key terms

• Quadratic function: A function that can be written as f(x) = ax² + bx + c, where a ≠ 0.
• Parabola: The graph of a quadratic function.
• x-intercept: A point where a graph meets the x-axis and y = 0.
• y-intercept: A point where a graph meets the y-axis and x = 0.
• Zero: An input value that makes a function equal to zero.
• Root: A solution of an equation; the real roots of f(x) = 0 give the x-coordinates of its x-intercepts.
• Vertex: The maximum or minimum point of a parabola.
• Axis of symmetry: The vertical line through the vertex that divides a parabola into matching halves.
• Repeated root: A root occurring twice in the factorization of a quadratic, giving one distinct x-intercept.
• Discriminant: The expression b² − 4ac, which determines the number of real roots of a quadratic equation.

Key takeaways

• To find a y-intercept, set x = 0.
• To find x-intercepts, set y = 0 and solve.
• The y-intercept of y = ax² + bx + c is (0, c).
• A quadratic can have two, one, or no x-intercepts.
• Factorization, square roots and the quadratic formula can be used to find x-intercepts.
• Intercepts, symmetry, opening direction and the vertex help you sketch a parabola.
• In applications, explain intercepts using the quantities and units shown on the axes.

4. Domain and Range

What are domain and range?

A function connects an input to an output. For a function written as y = f(x), the input is x and the output is y.

The domain is the set of all allowed input values.

The range is the set of all output values the function actually produces from those inputs.

For example, consider:

y = x²

We can substitute any real number for x:

• If x = −3, then y = 9.
• If x = 0, then y = 0.
• If x = 2.5, then y = 6.25.

However, squaring a real number never produces a negative result.

Therefore:

• Domain: All real numbers.
• Range: All real numbers greater than or equal to zero.

The domain and range describe different features of the same function. A function can accept negative inputs without producing negative outputs.

Identifying domain and range from a graph

When examining a graph:

• Read horizontally to identify the domain: which x-values have points on the graph?
• Read vertically to identify the range: which y-values occur on the graph?

Imagine projecting the entire graph onto each axis. Its horizontal coverage gives the domain, and its vertical coverage gives the range.

For a complete parabola representing y = ax² + bx + c:

• The graph continues indefinitely to the left and right.
• Its vertical extent is limited by a minimum or maximum at the vertex.

The edge of a graphing window is not automatically the end of the function. Unless a restriction is stated or endpoints are marked, a quadratic graph continues beyond the displayed area.

 

SeThe first two graphs represent complete quadratic functions, continuing beyond the viewing window. The third show onlythe interval during which the ball is in flight. The dashed horizontal lines mark the vertex’s output value

The domain of a quadratic function

A quadratic function can be written as:

f(x) = ax² + bx + c, where a ≠ 0

Squaring, multiplying and adding are defined for every real value of x. Therefore, the natural domain of a quadratic function is all real numbers.

This includes:

• Positive and negative integers.
• Zero.
• Fractions and terminating or recurring decimals.
• Irrational numbers, such as √2 and π.

We can write this as:

x ∈ ℝ

This means “x belongs to the set of real numbers.”

In interval notation:

Domain: (−∞, ∞)

Worked example

Identify the domain of:

f(x) = −3x² + 6x − 8

There are no restrictions on substituting real values for x.

Domain: (−∞, ∞)

The negative coefficient does not restrict the domain. It changes the direction in which the parabola opens.

A question or real-world situation may specify a smaller domain. Always check for such restrictions before giving an answer.

How the vertex determines the range

The vertex is the turning point of a parabola. Its y-coordinate gives the minimum or maximum output of an unrestricted quadratic function.

If the vertex is (h, k):

• An upward-opening parabola has a minimum value of k.
• A downward-opening parabola has a maximum value of k.

The x-coordinate h tells us where the turning point occurs. The y-coordinate k tells us the limiting output value.

These rules assume that the full quadratic domain is being used.

Example: an upward-opening parabola

For:

y = (x − 2)² − 3

The vertex is (2, −3), and the parabola opens upwards.

The lowest output is −3. Every larger output is possible.

Range: y ≥ −3, or [−3, ∞).

Example: a downward-opening parabola

For:

y = −(x + 1)² + 5

The vertex is (−1, 5), and the parabola opens downwards.

The highest output is 5. Every smaller output is possible.

Range: y ≤ 5, or (−∞, 5].

Understanding interval notation

Interval notation describes a continuous set of numbers using its boundaries.

• A square bracket, [ or ], means the endpoint is included.
• A round bracket, ( or ), means the endpoint is excluded.
• The symbol ∞ means the interval extends without an upper bound.
• The symbol −∞ means the interval extends without a lower bound.

Infinity is not a number that can be reached, so always use round brackets with infinity.

The notation [0, 4] represents every real number between 0 and 4, including the endpoints. It does not mean only the whole numbers 0, 1, 2, 3 and 4.

Finding the range from vertex form

The vertex form of a quadratic is:

y = a(x − h)² + k

This form reveals:

• The vertex: (h, k).
• The opening direction: determined by the sign of a.
• The minimum or maximum output: k.

Worked example

Find the domain and range of:

y = 2(x − 4)² − 7

The vertex is (4, −7).

Because a = 2 is positive, the parabola opens upwards.

A squared expression is never negative:

(x − 4)² ≥ 0

Therefore:

2(x − 4)² − 7 ≥ −7

The minimum output is −7, reached when x = 4.

Domain: (−∞, ∞)
Range: [−7, ∞)

Worked example

Find the domain and range of:

y = −3(x + 2)² + 6

Rewrite x + 2 mentally as x − (−2). The vertex is (−2, 6).

Because a = −3 is negative, the parabola opens downwards. Its maximum output is 6.

Domain: (−∞, ∞)
Range: (−∞, 6]

Finding the range from expanded form

When a quadratic is written as:

y = ax² + bx + c

the vertex may not be obvious.

Find the vertex’s x-coordinate using:

x = −b / (2a)

Then substitute this value into the equation to find its y-coordinate.

Worked example

Find the domain and range of:

y = x² − 6x + 5

Identify the coefficients:

a = 1, b = −6, c = 5

Find the vertex’s x-coordinate:

x = −(−6) / (2 × 1)
x = 3

Substitute x = 3:

y = 3² − 6(3) + 5
y = 9 − 18 + 5
y = −4

The vertex is (3, −4).

Because a is positive, the vertex is a minimum.

Domain: (−∞, ∞)
Range: [−4, ∞)

Notice that the constant term, 5, gives the y-intercept, not the minimum value.

How changes to a quadratic affect its range

Start with the parent function:

y = x²

Its range is [0, ∞).

Moving the graph vertically

For y = x² + 4, every output increases by 4.

The minimum becomes 4, so the range is [4, ∞).

For y = x² − 6, every output decreases by 6.

The minimum becomes −6, so the range is [−6, ∞).

Moving the graph horizontally

For y = (x − 3)², the vertex moves to (3, 0).

The minimum remains zero, so the range is still [0, ∞).

A horizontal shift alone does not change the range of a complete parabola.

Reflecting the graph

For y = −x², the graph opens downwards.

Its maximum is zero, so the range is (−∞, 0].

Changing the width

The functions y = x² and y = 4x² have different widths but the same minimum value.

Both have range [0, ∞).

Restricted domains change the range

If only some inputs are allowed, only the outputs produced by those inputs belong to the range.

Consider:

y = x², with −1 ≤ x ≤ 3

The domain is [−1, 3].

Evaluate the endpoints:

• At x = −1, y = 1.
• At x = 3, y = 9.

The vertex at x = 0 lies inside the domain and gives y = 0.

Therefore:

Range: [0, 9]

Looking only at the endpoints would miss the minimum.

Now consider the same equation with:

2 ≤ x ≤ 3

The vertex is outside this domain. Over the allowed interval, x² increases from 4 to 9.

Domain: [2, 3]
Range: [4, 9]

For a quadratic on a closed interval, check:

• The output at each endpoint.
• The output at the vertex, if its x-coordinate lies within the interval.

The smallest and largest of these outputs determine the range.

Domain and range in real-world situations

An equation may accept every real number mathematically, while only some inputs make sense in context.

Time, length and the number of objects often have practical restrictions.

Worked example: a ball in flight

A simplified model describes a ball’s height:

h = −5t² + 20t

Here:

• t is time in seconds after launch.
• h is height in metres above the ground.

The model describes the flight from launch until the ball returns to the ground.

Find the practical domain.

At ground level, h = 0:

−5t² + 20t = 0
−5t(t − 4) = 0

Therefore:

t = 0 or t = 4

The ball launches at 0 seconds and lands at 4 seconds.

Domain: [0, 4] seconds

Find the practical range.

The parabola opens downwards, so its vertex gives the maximum height.

t = −20 / [2(−5)]
t = 2

Substitute:

h = −5(2)² + 20(2)
h = −20 + 40
h = 20

The ball reaches 20 metres and is never below ground during the modelled flight.

Range: [0, 20] metres

The complete mathematical parabola has range (−∞, 20], but the flight model uses only the portion from launch to landing.

Continuous and discrete domains

Not every real-world domain is a continuous interval.

Continuous quantities, such as time and length, can take any real value within an interval.

Discrete quantities, such as numbers of tickets or students, take separate values.

For example, if n represents the number of tickets sold for an event with 100 seats, the domain is:

{0, 1, 2, …, 100}

Writing [0, 100] alone would incorrectly include values such as 2.5 tickets.

If a quadratic model uses this discrete domain, its range consists of the outputs for those permitted whole-number inputs. It does not necessarily include every value between its minimum and maximum.

Common misconceptions

• “Domain means the y-values.” Domain describes inputs; range describes outputs.
• “A quadratic cannot have negative inputs because it contains x².” Negative numbers can be squared.
• “Every quadratic has range y ≥ 0.” The range depends on the vertex, opening direction and any domain restriction.
• “The vertex’s x-coordinate gives the range boundary.” The y-coordinate gives the minimum or maximum output.
• “Infinity should have a square bracket.” Infinity is never included as an endpoint.
• “The graph ends at the edge of the image.” A viewing window shows only part of an unrestricted parabola.
• “Restricting the domain leaves the range unchanged.” Removing inputs can also remove possible outputs.

Did you know?

Every output above the minimum of a complete upward-opening parabola occurs at two different inputs, one on each side of the axis of symmetry.

The minimum output occurs at only one input: the vertex’s x-coordinate.

For example, in y = (x − 2)² − 3:

• y = 1 occurs at x = 0 and x = 4.
• y = −3 occurs only at x = 2.

Key terms

• Domain: The set of all allowed input values of a function.
• Range: The set of all outputs produced by the allowed inputs.
• Real numbers: Numbers on the number line, including rational and irrational numbers.
• Vertex: The turning point of a parabola.
• Minimum value: The smallest output a function reaches.
• Maximum value: The largest output a function reaches.
• Interval notation: A way to describe a continuous set of numbers using boundaries and brackets.
• Restricted domain: A domain limited by a stated condition or real-world context.
• Continuous domain: A domain containing every real value within an interval.
• Discrete domain: A domain consisting of separate permitted values.

Key takeaways

• The domain describes inputs, while the range describes outputs.
• An unrestricted quadratic function has domain (−∞, ∞).
• For vertex (h, k), an upward-opening parabola has range [k, ∞).
• For vertex (h, k), a downward-opening parabola has range (−∞, k].
• Square brackets include endpoints; round brackets exclude them.
• Always use round brackets with infinity.
• For restricted domains, check endpoints and any vertex within the domain.
• In real-world models, interpret domain and range using appropriate quantities, units and practical restrictions.

5. Graphing Quadratic Functions

Why graph a quadratic function?

An equation describes a relationship using algebra. A graph makes that relationship visible.

Graphing a quadratic helps us identify:

• Where the function increases or decreases.
• Its maximum or minimum value.
• Where the graph meets the coordinate axes.
• Its symmetry.
• Its domain and range.

A quadratic function has the form:

y = ax² + bx + c, where a ≠ 0

Its graph is a parabola: a smooth curve with one turning point.

• If a > 0, the parabola opens upwards.
• If a < 0, the parabola opens downwards.

A table of values provides points on the curve. The equation’s other features help us connect those points correctly.

Creating a table of values

A table of values lists selected inputs and their corresponding outputs.

To create one:

1. Choose suitable x-values.
2. Substitute each x-value into the equation.
3. Calculate the corresponding y-value.
4. Record each pair as a point (x, y).

Choose values on both sides of the vertex whenever possible. This shows both branches of the parabola and makes its symmetry easier to recognize.

Worked example

Create a table for:

y = x² − 4x + 1

For this quadratic, the vertex’s x-coordinate is:

x = −b / (2a)
x = −(−4) / (2 × 1)
x = 2

Choose x-values from −1 to 5, centred around x = 2.

The points are:

(−1, 6), (0, 1), (1, −2), (2, −3), (3, −2), (4, 1), (5, 6)

Notice the matching outputs on either side of x = 2:

• x = 1 and x = 3 both give y = −2.
• x = 0 and x = 4 both give y = 1.
• x = −1 and x = 5 both give y = 6.

These pairs reveal the graph’s symmetry.

Substituting negative numbers correctly

Negative inputs are a common source of calculation errors. Use brackets when substituting them.

For x = −1:

y = (−1)² − 4(−1) + 1
y = 1 + 4 + 1
y = 6

Two important rules are involved:

• Squaring a negative number gives a positive result: (−1)² = 1.
• Multiplying two negative numbers gives a positive result: −4 × −1 = 4.

Also distinguish between:

(−2)² = 4

and:

−2² = −4

In the second expression, the exponent is applied before the negative sign. On a calculator, brackets make your intended input clear.

Choosing an appropriate scale

Before plotting, inspect the smallest and largest values in your table.

For the example:

• x-values run from −1 to 5.
• y-values run from −3 to 6.

Choose axes that comfortably include these points and leave space around the vertex.

A clear graph should have:

• A labelled horizontal x-axis and vertical y-axis.
• Evenly spaced numerical intervals on each axis.
• A scale that uses the available space effectively.
• Units where quantities represent measurements.
• The function’s equation as a title or label.

The two axes may use different scales. However, the scale must remain consistent along each individual axis.

For example, one square could represent 1 unit horizontally and 2 units vertically. Label this clearly.

Plotting points accurately

Coordinates are written in the order:

(x, y)

Start at the origin, move horizontally according to x, and then vertically according to y.

For example:

• (−1, 6): Move 1 unit left and 6 units up.
• (1, −2): Move 1 unit right and 2 units down.
• (2, −3): Move 2 units right and 3 units down.

Use small dots or crosses so that the position of each point is clear.

After plotting, compare each point with its table entry. Reversing the coordinates of a point changes its location.

The left graph shows the table values. The right graph shows the smooth parabola through them, with its vertex in pink, axis of symmetry dashed, and x-intercepts in orange.

Drawing the parabola

Once the points are plotted, draw a smooth curve through them.

A quadratic graph should:

• Have one rounded turning point.
• Be symmetrical about a vertical line.
• Pass through the correctly calculated points.
• Continue smoothly on both sides.

Do not join consecutive points with straight line segments. This produces corners that do not belong to a parabola.

The table contains only a selection of points. The curve also contains the points for inputs between those listed.

For example, when x = 1.5:

y = 1.5² − 4(1.5) + 1
y = 2.25 − 6 + 1
y = −2.75

Therefore, (1.5, −2.75) lies on the curve between the plotted points (1, −2) and (2, −3).

Unless a domain restriction is given, continue the curve towards the edges of the graph and use arrows to indicate that it extends further.

Identifying important features

For the function:

y = x² − 4x + 1

we can identify several features while graphing.

Opening direction

The coefficient of x² is positive, so the parabola opens upwards.

Vertex

The lowest point is:

(2, −3)

This is the vertex, and the minimum output is −3.

Axis of symmetry

The vertical line through the vertex is:

x = 2

Write the axis of symmetry as an equation, rather than simply “2.”

y-intercept

Set x = 0:

y = 1

The y-intercept is (0, 1).

x-intercepts

Set y = 0:

x² − 4x + 1 = 0

Using the quadratic formula:

x = [4 ± √(16 − 4)] / 2
x = 2 ± √3

The x-intercepts are approximately:

(0.268, 0) and (3.732, 0)

These intercepts occur between the integer inputs in the table. A table does not need to contain an output of exactly zero for the graph to cross the x-axis.

Domain and range

For the complete quadratic:

• Domain: (−∞, ∞)
• Range: [−3, ∞)

Using symmetry to make graphing more efficient

Points the same horizontal distance from the axis of symmetry have the same y-coordinate.

For a parabola with axis x = 2:

• The reflection of (0, 1) is (4, 1).
• The reflection of (1, −2) is (3, −2).
• The reflection of (−1, 6) is (5, 6).

Once you know the axis of symmetry, you can calculate points on one side and reflect them to the other.

This also provides a useful check. If two inputs are equally far from the symmetry axis but your calculated outputs differ, recheck your arithmetic.

Graphing a downward-opening quadratic

Worked example

Sketch:

y = −x² + 2x + 3

Find the vertex’s x-coordinate:

x = −2 / [2(−1)]
x = 1

Choose inputs around x = 1.

The table reveals:

• Vertex: (1, 4).
• Axis of symmetry: x = 1.
• y-intercept: (0, 3).
• x-intercepts: (−1, 0) and (3, 0).

Because the coefficient of x² is negative, the graph opens downwards. Draw a smooth curve rising to the maximum at (1, 4), then falling symmetrically.

For the complete function:

• Domain: (−∞, ∞)
• Range: (−∞, 4]

When integer inputs miss the vertex

A table using only whole-number inputs may not include the exact turning point.

Consider:

y = x² − 3x + 1

The vertex’s x-coordinate is:

x = −(−3) / (2 × 1)
x = 1.5

A table gives y = −1 at both x = 1 and x = 2. Neither point is the vertex.

Calculate the output halfway between them:

y = 1.5² − 3(1.5) + 1
y = 2.25 − 4.5 + 1
y = −1.25

The vertex is (1.5, −1.25).

Add this point to the table before drawing the curve. Otherwise, you might accidentally draw a flat bottom or place the minimum too high.

Checking whether a graph is reasonable

A finished graph should agree with both the table and the equation.

Check the opening direction.

• Positive a: opens upwards.
• Negative a: opens downwards.

Check the y-intercept.

For y = ax² + bx + c, the graph must pass through (0, c).

Check the vertex and symmetry.

The turning point should lie on:

x = −b / (2a)

The two sides should be mirror images across this line.

Check the intercepts.

Any calculated real roots should match the points where the curve meets the x-axis.

Check an additional point.

Choose an input not used in the original table, calculate its output and see whether it lies on the drawn curve.

Check the shape.

A quadratic should have a smooth turn. It should not contain corners, loops or several turning points.

Check the domain and context.

If the question restricts the inputs, show only the required portion of the graph.

Example of spotting an error

A student sketches y = x² − 4x + 1 opening downwards with vertex (2, 3).

Two checks reveal problems:

• The positive coefficient of x² requires an upward-opening curve.
• Substituting x = 2 gives y = −3, so the vertex should be (2, −3).

These checks identify the errors without recalculating the entire table.

Using graphing technology

A graphing calculator or graphing application can help verify a hand-drawn graph.

Enter the equation carefully, then choose a viewing window that includes:

• The vertex.
• The y-intercept where useful.
• Any x-intercepts.
• Enough of both branches to show the shape.

A poor viewing window can hide important features. For example, a window showing only one branch may make a parabola appear almost straight.

Use the table or point-evaluation feature to compare several outputs with your calculations. Technology is most useful when you can explain why the displayed graph makes sense.

Real-world connection: graphing a ball’s flight

A simplified model gives the height of a ball as:

h = −5t² + 20t

Here, h is height in metres and t is time in seconds after launch.

Plot time on the horizontal axis and height on the vertical axis.

The graph shows:

• The ball starts at ground level.
• It reaches a maximum height of 20 m after 2 s.
• It returns to ground level after 4 s.

For the flight, use the domain 0 ≤ t ≤ 4. The modelled graph stops at landing.

This is a height–time graph. It shows how height changes with time; it does not show the ball’s physical path through space.

Did you know?

For equally spaced x-values, a quadratic table has constant second differences.

For y = x² − 4x + 1, the outputs in our first table are:

6, 1, −2, −3, −2, 1, 6

Subtract consecutive outputs to find the first differences:

−5, −3, −1, 1, 3, 5

Subtract consecutive first differences:

2, 2, 2, 2, 2

The constant second difference is a useful pattern for checking a table. For x-values spaced 1 unit apart, it equals 2a.

Common misconceptions

• “A parabola is made from straight segments.” Draw a smooth curve through the points.
• “The smallest output in my table must be the minimum.” The vertex may lie between the selected inputs.
• “Every parabola is symmetrical about the y-axis.” Its axis is x = −b / (2a), which may be elsewhere.
• “No zero in the table means no x-intercepts.” Intercepts may occur between the listed inputs.
• “Five points are always enough.” Add points when needed to locate the vertex, intercepts or uncertain parts of the curve.
• “A calculator graph must show every important feature.” The viewing window may hide them.

Key terms

• Table of values: A list of selected inputs and their corresponding outputs.
• Ordered pair: A point written as (x, y).
• Scale: The numerical value represented by intervals on an axis.
• Parabola: The smooth curve representing a quadratic function.
• Vertex: The maximum or minimum point of a parabola.
• Axis of symmetry: The vertical line dividing a parabola into matching halves.
• Intercept: A point where a graph meets a coordinate axis.
• Sketch: A drawing showing a graph’s overall shape and important features.
• Viewing window: The portion of the coordinate plane displayed by graphing technology.
• Second differences: Differences between consecutive first differences in a sequence of outputs.

Key takeaways

• Create a table by substituting carefully chosen inputs into the equation.
• Include values on both sides of the vertex.
• Use brackets when substituting negative numbers.
• Label axes and use consistent scales.
• Plot coordinates accurately and draw a smooth curve.
• Identify the vertex, symmetry axis, intercepts and opening direction.
• Add extra points when the original table misses an important feature.
• Verify the graph against the equation, and respect any domain restrictions.