Understanding Quadratic Functions

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.

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.

---

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

---

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.

---

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 

In each case, the coefficient of x² is positive.

---

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 = −7x² + 2x − 1

In each case, the coefficient of x² is negative.

Quick Rule

---

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

---

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​

---

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.

---

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?

---

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

---

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

---

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.

---

Parabolas in Real-World Situations

Parabolic shapes and quadratic models appear in many situations.

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.

---

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.

---

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.