Understanding Quadratic Functions

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.

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.