Quadratic graphs

​​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.

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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.

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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 $$\underline{(-2,0)}$$​​

​​Drawing a quadratic graph

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

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 $$ax^2+bx+c=0$$.