17.2 What is ‘a’ in Quadratic Equation?

17.2 What is ‘a’ in Quadratic Equation?

In a quadratic equation,$$y=ax^2+bx+c$$

the a is the coefficient of $x^2$. It is especially important because it controls the direction and shape of the parabola.

If:$$a>0$$

the curve opens upwards.

If:$$a<0$$

the curve opens downwards.

Also, the larger $|a|$ is, the narrower the curve becomes.

How can we find $a$?

For O Level work, there are 4 common ways.

1. Read it directly from the equation

Example:$$y=3x^2-4x+7$$

The coefficient of $x^2$ is 3.

So:$$a=3$$

Another example:$$y=-2x^2+5x-1$$

So:$$a=-2$$

This is the easiest case.

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2. Use the turning point and another point

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

use the vertex form:$$y=a(x-h)^2+k$$

Suppose the turning point is:$$(2,-1)$$

and the graph passes through:$$(0,3)$$

Then:$$y=a(x-2)^2-1$$

Substitute $(0,3)$:$$3=a(0-2)^2-1$$$$3=4a-1$$$$4=4a$$$$a=1$$

So the equation begins:$$y=(x-2)^2-1$$

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3. Use the roots and another point

If the roots are $r_1$ and $r_2$, use:$$y=a(x-r_1)(x-r_2)$$

Suppose the roots are:$$x=1$$

and$$x=3$$

Then:$$y=a(x-1)(x-3)$$

If the graph passes through:$$(0,3)$$

substitute:$$3=a(0-1)(0-3)$$$$3=3a$$

Therefore:$$a=1$$

So:$$y=(x-1)(x-3)$$

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4. Use three points on the graph

If you know three points, substitute them into:$$y=ax^2+bx+c$$

Suppose the points are:$$(0,3),\quad(1,0),\quad(2,-1)$$

Using $(0,3)$:$$3=a(0)^2+b(0)+c$$

So:$$c=3$$

Using $(1,0)$:$$0=a+b+3$$

Using $(2,-1)$:$$-1=4a+2b+3$$

You then solve the simultaneous equations to find $a$ and $b$.

In this example:$$a=1$$

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⭐ Quick Summary

There are 4 main ways to find $a$:

Information given	Method
Full quadratic equation	Read coefficient of $x^2$
Turning point + another point	Use $y=a(x-h)^2+k$
Two roots + another point	Use $y=a(x-r_1)(x-r_2)$
Three points	Use $y=ax^2+bx+c$ and simultaneous equations

Remember

$$a>0 \Rightarrow \text{opens upwards}$$$$a<0 \Rightarrow \text{opens downwards}$$

And $a$ also controls how wide or narrow the parabola is.