17.10 Turning Point of a Quadratic Curve

🟦 Concept

A turning point is the point on a quadratic curve where the curve changes direction.

Look at the two quadratic curves in the diagram.

The curve on the left moves downwards, reaches its lowest point, and then starts moving upwards. This lowest point is called a minimum turning point.

The curve on the right moves upwards, reaches its highest point, and then starts moving downwards. This highest point is called a maximum turning point.

Every quadratic curve has one turning point.

If the turning point is written as$$(h,k)$$

then:

• $h$ is the x-coordinate of the turning point.
• $k$ is the y-coordinate of the turning point.

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🟨 Minimum and Maximum Turning Points

There are two types of turning points.

Minimum Turning Point

When a quadratic curve opens upwards, like ∪, its turning point is the lowest point on the curve.

Think of it as the bottom of a valley.

In the left-hand graph, $(h,k)$ represents the minimum turning point.

Maximum Turning Point

When a quadratic curve opens downwards, like ∩, its turning point is the highest point on the curve.

Think of it as the top of a hill.

In the right-hand graph, $(h,k)$ represents the maximum turning point.

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🟩 Formula

A quadratic equation can be written in the form$$y=ax^2+bx+c$$

The x-coordinate of its turning point can be found using$$x=-\frac{b}{2a}$$

After finding $x$, substitute this value into the original equation to find $y$.

The turning point is then written as$$(x,y)$$

Remember

Step 1: Find $a$ and $b$.

Step 2: Calculate$$x=-\frac{b}{2a}$$

Step 3: Substitute the value of $x$ into the original equation.

Step 4: Find $y$.

Step 5: Write the turning point as$$(x,y)$$

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🟪 Example Math

Find the turning point of$$y=x^2-4x+3$$

Step 1: Identify $a$ and $b$

Compare$$y=x^2-4x+3$$

with$$y=ax^2+bx+c$$

Therefore,$$a=1$$

and$$b=-4$$

Step 2: Find the x-coordinate

Use$$x=-\frac{b}{2a}$$

Substitute $a=1$ and $b=-4$:$$x=-\frac{-4}{2(1)}$$$$x=\frac{4}{2}$$$$x=2$$

So the x-coordinate of the turning point is 2.

Step 3: Find the y-coordinate

Substitute $x=2$ into the original equation:$$y=x^2-4x+3$$$$y=(2)^2-4(2)+3$$$$y=4-8+3$$$$y=-1$$

Therefore, the turning point is$$(2,-1)$$

Because the curve opens upwards, $(2,-1)$ is a minimum turning point.

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🟦 Line of Symmetry

Look again at the dashed vertical line in the reference diagram.

This is called the line of symmetry.

It passes directly through the turning point and divides the quadratic curve into two matching halves.

If the turning point is$$(h,k)$$

then the line of symmetry is$$x=h$$

For example, our turning point was$$(2,-1)$$

Therefore, the line of symmetry is$$x=2$$

Notice something important:

The line of symmetry uses only the x-coordinate of the turning point.

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💡 Exam Tip

If Cambridge gives you a quadratic graph, you may be able to find the turning point simply by reading it from the graph.

Find the point where the curve stops moving in one direction and begins moving in the other direction.

For example, suppose the lowest point of a quadratic curve is$$(3,-4)$$

Then its turning point is simply$$(3,-4)$$

and its line of symmetry is$$x=3$$

Always write coordinates in the correct order:$$(x,y)$$

x first, y second.

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⚠️ Common Mistake

Do not confuse the turning point with the roots of a quadratic graph.

Turning Point

The turning point is where the curve changes direction.

Roots

The roots are the points where the quadratic curve crosses or touches the x-axis.

At a root,$$y=0$$

For example, a quadratic curve might have two roots, but it still has only one turning point.

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❓ Question & Answer

1. What is a turning point?

A turning point is the point where a quadratic curve changes direction.

2. How many turning points does a quadratic curve have?

A quadratic curve has one turning point.

3. What is a minimum turning point?

It is the lowest point of a quadratic curve that opens upwards.

4. What is a maximum turning point?

It is the highest point of a quadratic curve that opens downwards.

5. What is the turning point written as?

It is written as a coordinate:$$(x,y)$$

or, in general,$$(h,k)$$

6. If the turning point is $(5,-3)$, what is the line of symmetry?

The x-coordinate is 5, so$$x=5$$

7. If the turning point is $(-2,6)$, what is the line of symmetry?

The x-coordinate is $-2$, so$$x=-2$$

8. How do I find the x-coordinate of the turning point from an equation?

For$$y=ax^2+bx+c$$

use$$x=-\frac{b}{2a}$$

Then substitute the value of $x$ back into the equation to find $y$.

9. Is a turning point the same as a root?

No.

A turning point is where the curve changes direction.

A root is where the curve meets the x-axis.

10. How can I tell whether the turning point is a maximum or minimum?

Look at the direction in which the parabola opens:

∪ opens upwards → Minimum

∩ opens downwards → Maximum

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📚 Want to Know More?

Once you understand turning points, continue with these closely related quadratic graph topics:

Quadratic Graphs → understand the overall shape of a parabola

Roots of a Quadratic Graph → find where the curve meets the x-axis

Line of Symmetry → understand how the parabola is divided into two equal halves

Domain of a Quadratic Graph → identify the possible x-values

Range of a Quadratic Graph → identify the possible y-values

Maximum and Minimum Values → use the turning point to determine the greatest or least value of a quadratic function

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⭐ Quick Summary

A quadratic curve has one turning point.

∪ Upward-opening curve → minimum turning point

∩ Downward-opening curve → maximum turning point

For$$y=ax^2+bx+c$$

the x-coordinate of the turning point is$$x=-\frac{b}{2a}$$

Substitute this value into the equation to find $y$.

If the turning point is$$(h,k)$$

then the line of symmetry is$$x=h$$

Turning point = the highest or lowest point where the quadratic curve changes direction.