# 0.6 Quadratic functions and graphs

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## Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of quadratic functions.

## Functions of the form $y=a{\left(x+p\right)}^{2}+q$

This form of the quadratic function is slightly more complex than the form studied in Grade 10, $y=a{x}^{2}+q$ . The general shape and position of the graph of the function of the form $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is shown in [link] .

## Investigation : functions of the form $y=a{\left(x+p\right)}^{2}+q$

1. On the same set of axes, plot the following graphs:
1. $a\left(x\right)={\left(x-2\right)}^{2}$
2. $b\left(x\right)={\left(x-1\right)}^{2}$
3. $c\left(x\right)={x}^{2}$
4. $d\left(x\right)={\left(x+1\right)}^{2}$
5. $e\left(x\right)={\left(x+2\right)}^{2}$
Use your results to deduce the effect of $p$ .
2. On the same set of axes, plot the following graphs:
1. $f\left(x\right)={\left(x-2\right)}^{2}+1$
2. $g\left(x\right)={\left(x-1\right)}^{2}+1$
3. $h\left(x\right)={x}^{2}+1$
4. $j\left(x\right)={\left(x+1\right)}^{2}+1$
5. $k\left(x\right)={\left(x+2\right)}^{2}+1$
Use your results to deduce the effect of $q$ .
3. Following the general method of the above activities, choose your own values of $p$ and $q$ to plot 5 different graphs (on the same set of axes) of $y=a{\left(x+p\right)}^{2}+q$ to deduce the effect of $a$ .

From your graphs, you should have found that $a$ affects whether the graph makes a smile or a frown. If $a<0$ , the graph makes a frown and if $a>0$ then the graph makes a smile. This was shown in Grade 10.

You should have also found that the value of $q$ affects whether the turning point of the graph is above the $x$ -axis ( $q<0$ ) or below the $x$ -axis ( $q>0$ ).

You should have also found that the value of $p$ affects whether the turning point is to the left of the $y$ -axis ( $p>0$ ) or to the right of the $y$ -axis ( $p<0$ ).

These different properties are summarised in [link] . The axes of symmetry for each graph is shown as a dashed line.

 $p<0$ $p>0$ $a>0$ $a<0$ $a>0$ $a<0$ $q\ge 0$ $q\le 0$

## Domain and range

For $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ , the domain is $\left\{x:x\in \mathbb{R}\right\}$ because there is no value of $x\in \mathbb{R}$ for which $f\left(x\right)$ is undefined.

The range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ depends on whether the value for $a$ is positive or negative. We will consider these two cases separately.

If $a>0$ then we have:

$\begin{array}{ccc}\hfill {\left(x+p\right)}^{2}& \ge & 0\phantom{\rule{1.em}{0ex}}\left(\mathrm{The square of an expression is always positive}\right)\hfill \\ \hfill a{\left(x+p\right)}^{2}& \ge & 0\phantom{\rule{1.em}{0ex}}\left(\mathrm{Multiplication by a positive number maintains the nature of the inequality}\right)\hfill \\ \hfill a{\left(x+p\right)}^{2}+q& \ge & q\hfill \\ \hfill f\left(x\right)& \ge & q\hfill \end{array}$

This tells us that for all values of $x$ , $f\left(x\right)$ is always greater than or equal to $q$ . Therefore if $a>0$ , the range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is $\left\{f\left(x\right):f\left(x\right)\in \left[q,\infty \right)\right\}$ .

Similarly, it can be shown that if $a<0$ that the range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is $\left\{f\left(x\right):f\left(x\right)\in \left(-\infty ,q\right]\right\}$ . This is left as an exercise.

For example, the domain of $g\left(x\right)={\left(x-1\right)}^{2}+2$ is $\left\{x:x\in \mathbb{R}\right\}$ because there is no value of $x\in \mathbb{R}$ for which $g\left(x\right)$ is undefined. The range of $g\left(x\right)$ can be calculated as follows:

$\begin{array}{ccc}\hfill {\left(x-p\right)}^{2}& \ge & 0\hfill \\ \hfill {\left(x+p\right)}^{2}+2& \ge & 2\hfill \\ \hfill g\left(x\right)& \ge & 2\hfill \end{array}$

Therefore the range is $\left\{g\left(x\right):g\left(x\right)\in \left[2,\infty \right)\right\}$ .

## Domain and range

1. Given the function $f\left(x\right)={\left(x-4\right)}^{2}-1$ . Give the range of $f\left(x\right)$ .
2. What is the domain of the equation $y=2{x}^{2}-5x-18$ ?

## Intercepts

For functions of the form, $y=a{\left(x+p\right)}^{2}+q$ , the details of calculating the intercepts with the $x$ and $y$ axes is given.

The $y$ -intercept is calculated as follows:

$\begin{array}{ccc}\hfill y& =& a{\left(x+p\right)}^{2}+q\hfill \\ \hfill {y}_{int}& =& a{\left(0+p\right)}^{2}+q\hfill \\ & =& a{p}^{2}+q\hfill \end{array}$

If $p=0$ , then ${y}_{int}=q$ .

For example, the $y$ -intercept of $g\left(x\right)={\left(x-1\right)}^{2}+2$ is given by setting $x=0$ to get:

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