17.8 How ‘a‘ Changes the Width of a Parabola

🟦 Concept

A quadratic equation can be written as:$$y=ax^2+bx+c$$

The coefficient $a$ affects two important features of the parabola:

• whether the curve opens upwards or downwards
• how wide or narrow the curve appears

For the width, we look at the size of a, or more precisely:$$|a|$$

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🟩 When $a=1$

Consider:$$y=x^2$$

This gives the basic parabola that we can use for comparison.$$a=1$$

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🟪 When $|a|>1$

Consider:$$y=2x^2$$

Here:$$a=2$$

Since:$$|a|>1$$

the parabola becomes narrower and steeper than:$$y=x^2$$

For example:$$y=3x^2$$

is even narrower than:$$y=2x^2$$

Rule

As $|a|$ becomes larger, the parabola becomes narrower.

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🟨 When $0<|a|<1$

Consider:$$y=\frac{1}{2}x^2$$

Here:$$a=\frac{1}{2}$$

Since:$$0<|a|<1$$

the parabola becomes wider and flatter than:$$y=x^2$$

For example:$$y=\frac{1}{4}x^2$$

is wider than:$$y=\frac{1}{2}x^2$$

Rule

As $|a|$ gets closer to zero, the parabola becomes wider.

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🟧 What If $a$ Is Negative?

The sign of $a$ determines the direction of the parabola.

If:$$a>0$$

the parabola opens upwards.

If:$$a<0$$

the parabola opens downwards.

However, the width depends on ∣a∣|a|.

For example:$$y=2x^2$$

and:$$y=-2x^2$$

have the same width.

The only difference is their direction.

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📘 Compare These Three Graphs

$$y=\frac{1}{2}x^2$$

→ Wider$$y=x^2$$

→ Basic width$$y=2x^2$$

→ Narrower

So:$$\frac{1}{2}<1<2$$

but for comparing the shape, think about the magnitude:$$|a|$$

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💡 Exam Tip

Do not look only at whether $a$ is positive or negative.

First ask:

What is the value of ∣a∣|a|?

Then use:$$|a|>1$$

→ narrower

and:$$0<|a|<1$$

→ wider

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⚠️ Common Mistake

Students sometimes think a negative value of $a$ automatically makes the parabola wider.

That is incorrect.

For example:$$a=2$$

and:$$a=-2$$

produce parabolas with the same width.

The negative sign only changes the direction of the curve.

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⭐ Quick Summary

For:$$y=ax^2+bx+c$$$$|a|>1$$

→ parabola is narrower$$|a|=1$$

→ basic parabola width$$0<|a|<1$$

→ parabola is wider

And:$$a>0$$

→ opens upwards$$a<0$$

→ opens downwards

Remember

The sign of a controls the direction.

The size of ∣a∣|a| controls how wide or narrow the parabola appears.