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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.
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] .
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.
$p<0$ | $p>0$ | |||
$a>0$ | $a<0$ | $a>0$ | $a<0$ | |
$q\ge 0$ | ||||
$q\le 0$ |
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 $\left\{g\right(x):g(x)\in [2,\infty \left)\right\}$ .
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:
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