Chapter Overview Maths

Types of numbers

Number calculations

Fractions, decimals and percentages

Algebraic manipulation

Formulae and equations

Straight line graphs

Other graphs

Ratio

Proportion

Rates of change

Shapes

Properties of shapes

Measures

Lines and angles

Drawing shapes

Trigonometry

Probability

Statistics

Maths 0%

Summary

## ​​In a nutshell

Quadratic graphs have equations of the form $y=ax^2+bx+c$ and, when plotted, are curved like a $\cup$ or like an $\cap$. Solving a quadratic equation using the corresponding quadratic graph is a case of identifying the $x$-intercepts.

## Prerequisites

This summary assumes you are comfortable with at least one of the following methods: factorising quadratics; completing the square of a quadratic;  using the quadratic formula.

## What do quadratic graphs look like?

Quadratic graphs have either a $\cup$-shape (a positive quadratic, on the left below) or a $\cap$-shape (a negative quadratic, on the right below):

## What do their equations look like?

Quadratic equations have the form ​

$y=ax^2+bx+c$​​

where $a$, $b$ and $c$ are constant numbers and $a$ is not $0$​.

If $a$ is positive, then the graph will be $\cup$-shaped. If $a$ is negative, then it will be $\cap$-shaped. This is a quick, first indication as to what your quadratic curve will look like based on its equation.

Note: Unlike with linear equations, there is not a constant value for the gradient. This is because the gradient is not constant on a quadratic curve.

For example:

$y=x^2+5x+6$​​

This is the equation of a quadratic graph, where $a=1$, $b=5$ and $c=6$

## Identifying the $x$-intercepts

When looking for $x$-intercepts on any graph, set $y$​ in the equation to $0$ and work out the corresponding $x$-value(s). Hence, to find the $x$-intercepts of the graph of $y=ax^2+bx+c$, you solve:

$0=ax^2+bx+c$​​

This can be achieved using one of the three methods outlined in the prerequisites at the top.

##### Example 1

Find the $x$-intercepts of the curve with equation

$y=x^2+5x+6$

First set $y=0$, then factorise:

$0=x^2+5x+6\\0=(x+3)(x+2)$

Hence

$x=-3$ and $x=-2$

These give the two $x$-intercepts:

$\underline{(-3,0)}$ and $\underline{(-2,0)}$​​

Without using graphing software, only a rough sketch of a quadratic curve is enough. There are important features of the graph that you can get right and use in your sketch:

• the shape: either $\cup$-shaped or $\cap$-shaped. This is based on the $a$-term in $y=ax^2+bx+c$;​
• the $y$-intercept: this is the $c$-term in $y=ax^2+bx+c$;
• the $x$-intercept(s): sometimes there are two, sometimes one and sometimes none. Solving $0=ax^2+bx+c$ find these.​

Once you have established these three characteristics of the quadratic curve, you can give a fairly accurate sketch of it.

#### PROCEDURE

 1 Ensure the equation is in the form $y=ax^2+bx+c$. Rearrange to this if necessary.​ 2 Identify the shape of the quadratic curve. If $a>0$ then it is $\cup$-shaped, if $a<0$, then it is $\cap$-shaped.​ 3 Identify the $y$-intercept. This is the $c$-term in the equation of the curve. Mark this point on: $(0,c)$. Your curve will pass through it.​ 4 By solving $ax^2+bx+c=0$ (either by factorising, completing the square or using the quadratic formula), identify the $x$-intercept(s). Mark however many there are (zero, one or two) on the $x$-axis. If there are no solutions and hence no $x$-intercepts, then you may have to use graph transformations to draw the curve. 5 Join the points in the correct curve.
##### Example 2

Draw the curve that has equation

$y=x^2+5x+6$

For this equation, $a=1$ which is positive, so this curve is $\cup$-shaped. Since $c=6$, the $y$-intercept is at $(0,6)$. Finally, as given in the example above, the $x$-intercepts are at $(-2,0)$ and $(-3,0)$. Combining these gives the following sketch:​

## Solving quadratic equations using a graph

To find the $x$-intercept(s) of a quadratic curve, you have to solve a quadratic equation. This works the other way too: if you have the graph of $y=ax^2+bx+c$​, then the $x$-intercepts give the solutions to the quadratic equation $ax^2+bx+c=0$.