0.6 Quadratic functions and graphs

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 $y=a{(x+p)}^2+q$

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 $f\left(x\right)=a{(x+p)}^2+q$ is shown in [link] .

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

1. On the same set of axes, plot the following graphs:
   1. $a\left(x\right)={(x-2)}^2$
   2. $b\left(x\right)={(x-1)}^2$
   3. $c\left(x\right)=x^2$
   4. $d\left(x\right)={(x+1)}^2$
   5. $e\left(x\right)={(x+2)}^2$
   Use your results to deduce the effect of $p$ .
2. On the same set of axes, plot the following graphs:
   1. $f\left(x\right)={(x-2)}^2+1$
   2. $g\left(x\right)={(x-1)}^2+1$
   3. $h\left(x\right)=x^2+1$
   4. $j\left(x\right)={(x+1)}^2+1$
   5. $k\left(x\right)={(x+2)}^2+1$
   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 $y=a{(x+p)}^2+q$ 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 $f\left(x\right)=a{(x+p)}^2+q$ , the domain is $\{x:x\in \mathbb{R}\}$ because there is no value of $x\in \mathbb{R}$ for which $f\left(x\right)$ is undefined.

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

If $a>0$ then we have:

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 $f\left(x\right)=a{(x+p)}^2+q$ is $\left\{f\right(x):f(x)\in [q,\infty \left)\right\}$ .

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

For example, the domain of $g\left(x\right)={(x-1)}^2+2$ is $\{x:x\in \mathbb{R}\}$ 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:

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

Domain and range

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

Intercepts

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

The $y$ -intercept is calculated as follows:

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

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