17.5 Standard Form vs Factorised Form vs Vertex Form

1. Standard Form

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

This equation can be used to find a, b, and c using the simultaneous equations method. If you know the coordinates of three points on the graph, substitute their $x$– and $y$-values into the quadratic equation to form three equations. Then solve these equations simultaneously to find the values of a, b, and c.

What the letters tell us

$$a=\text{controls direction and width of the parabola}$$$$b=\text{helps determine the turning point and line of symmetry}$$$$c=\text{y-intercept}$$

y-intercept

$$(0,c)$$

Line of symmetry

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

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2. Factorised Form

$$y=a(x-r_1)(x-r_2)$$

where $r_1$ and $r_2$ are the roots.

Roots

$$x=r_1$$$$x=r_2$$

Therefore, the x-intercepts are:$$(r_1,0)$$

and$$(r_2,0)$$

Line of symmetry

The line of symmetry lies halfway between the roots:$$x=\frac{r_1+r_2}{2}$$

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3. Vertex Form

$$y=a(x-h)^2+k$$

Turning point

$$(h,k)$$

Line of symmetry

$$x=h$$

Minimum or maximum value

If:$$a>0$$

the parabola opens upwards and:$$y_{\min}=k$$

If:$$a<0$$

the parabola opens downwards and:$$y_{\max}=k$$

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Comparing the Three Forms

Form	Formula	Information seen directly
Standard Form	$\displaystyle y=ax^2+bx+c$	$a$, $b$, y-intercept $c$
Factorised Form	$\displaystyle y=a(x-r_1)(x-r_2)$	roots $r_1,r_2$
Vertex Form	$\displaystyle y=a(x-h)^2+k$	turning point $(h,k)$

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Same Quadratic in All Three Forms

Consider:$$y=x^2-4x+3$$

Standard Form

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

Factorised Form

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

Vertex Form

$$y=(x-2)^2-1$$

All three equations describe exactly the same parabola.

They simply show different information more clearly:$$\text{Standard Form}\rightarrow \text{y-intercept}$$$$\text{Factorised Form}\rightarrow \text{roots}$$$$\text{Vertex Form}\rightarrow \text{turning point}$$

Quick Formula Summary

$$y=ax^2+bx+c$$$$y=a(x-r_1)(x-r_2)$$$$y=a(x-h)^2+k$$

Same quadratic, different forms, different useful information.