5.2: Transformation of Graphs

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Page ID: 48976

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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For a given function, we now study how the graph of the function changes when performing elementary operations, such as adding, subtracting, or multiplying a constant number to the input or output. We will study the behavior in five specific examples.

Consider the following graphs:

$clipboard_eacf01f3495a998123b32303706956032.png$

We see that the function $y=x^2$ is shifted up by $2$ units, respectively down by $2$ units. In general, we have:

Observation: Shift-Up-Down

Consider the graph of a function $y=f(x)$. Then, the graph of $y=f(x)+c$ is that of $y=f(x)$ shifted up or down by $c$. If $c$ is positive, the graph is shifted up, if $c$ is negative, the graph is shifted down.

Next, we consider the transformation of $y=x^2$ given by adding or subtracting a constant to the input $x$.

$clipboard_e83d4f375b3adfb63d425181e07e22049.png$

Now, we see that the function is shifted to the left or right. Note, that $y=(x+1)^2$ shifts the function to the left, which can be seen to be correct, since the input $x=-1$ gives the output $y=((-1)+1)^2=0^2=0$.

Observation: Shift-Left-Right

Consider the graph of a function $y=f(x)$. Then, the graph of $y=f(x+c)$ is that of $y=f(x)$ shifted to the left or right by $c$. If $c$ is positive, the graph is shifted to the left, if $c$ is negative, the graph is shifted to the right.

Another transformation is given by multiplying the function by a fixed positive factor.

$clipboard_e5aee86938c686780f5d680badb76dded.png$

This time, the function is stretched away or compressed towards the $x$-axis.

Observation: Stretch

Consider the graph of a function $y=f(x)$ and let $c>0$. Then, the graph of $y=c\cdot f(x)$ is that of $y=f(x)$ stretched away or compressed towards the $x$-axis by a factor $c$. If $c>1$, the graph is stretched away from the $x$-axis, if $0<c<1$, the graph is compressed towards the $x$-axis.

Similarly, we can multiply the input by a positive factor.

$clipboard_e1f150aca6544bfc81ba51b6da979f17f.png$

This time, the function is stretched away or compressed towards the $y$-axis.

Observation: Stretch

Consider the graph of a function $y=f(x)$ and let $c>0$. Then, the graph of $y=f(c\cdot x)$ is that of $y=f(x)$ stretched away or compressed towards the $y$-axis by a factor $c$. If $c>1$, the graph is compressed towards the $y$-axis, if $0<c<1$, the graph is stretched away from the $y$-axis.

The last transformation is given by multiplying $(-1)$ to the input or output, as displayed in the following chart.

$clipboard_ed8d387a439315d49d4d2c54399f26df5.png$

Here, the function is reflected either about the $x$-axis or the $y$-axis.

Observation: Reflect

Consider the graph of a function $y=f(x)$. Then, the graph of $y=-f(x)$ is that of $y=f(x)$ reflected about the $x$-axis. Furthermore, the graph of $y=f(-x)$ is that of $y=f(x)$ reflected about the $y$-axis.

Example $\PageIndex{1}$

Guess the formula for the function, based on the basic graphs in Section 5.1 and the transformations described above.

$clipboard_e33ea7738f147ef2725d16e5d7b2055b1.png$
$clipboard_eb902a566987864008a7aac201e7490be.png$
$clipboard_e6b8a32eaf1022662d6f1fe968fe73b15.png$
$clipboard_e9bd4dcc934a6595ce89ecc341eecc85e.png$

Solution

This is the square-root function shifted to the left by $2$. Thus, by Observation, this is the function $f(x)=\sqrt{x+2}$.
This is the graph of $y=\dfrac 1 x$ reflected about the $x$-axis (or also $y=\dfrac 1 x$ reflected about the $y$-axis). In either case, we obtain the rule $y=-\dfrac 1 x$.
This is a parabola reflected about the $x$-axis and then shifted up by $3$. Thus, we get: \[\begin{aligned}
&y=x^{2} \\
\text{reflecting about the x-axis gives }&y=-x^{2} \\
\text{shifting the graph up by 3 gives }&y=-x^{2}+3
\end{aligned} \nonumber \]
Starting from the graph of the cubic equation $y=x^3$, we need to reflect about the $x$-axis (or also $y$-axis), then shift up by $2$ and to the right by $3$. These transformations affect the formula as follows: \[\begin{aligned}
& y=x^{3} \\
\text {reflecting about the x-axis gives } & y=-x^{3}\\
\text{shifting up by 2 gives } & y=-x^{3}+2\\
\text{shifting the the right by 3 gives } & y=-(x-3)^{3}+2
\end{aligned} \nonumber \]

All these answers can be checked by graphing the function with the TI-84.

Example $\PageIndex{2}$

Sketch the graph of the function, based on the basic graphs in Section 5.1 and the transformations described above.

$y=x^2+3$
$y=(x+2)^2$
$y=|x-3|-2$
$y=2\cdot \sqrt{x+1}$
$y=-\left(\dfrac 1 {x}+2\right)$
$y=(-x+1)^3$

Solution

This is the parabola $y=x^2$ shifted up by $3$. The graph is shown below.

$clipboard_ebd9e3d479e36e62ef33c44f173bc0f7e.png$

$y=(x+2)^2$ is the parabola $y=x^2$ shifted $2$ units to the left.
$clipboard_e84abcdd0f0f0b966023ac8e58ed9ac90.png$
The graph of the function $f(x)=|x-3|-2$ is the absolute value shifted to the right by $3$ and down by $2$. (Alternatively, we can first shift down by $2$ and then to the right by $3$.)

$clipboard_e4a44de08a35dd6eab8e06c99bbd34b72.png$

Similarly, to get from the graph of $y=\sqrt{x}$ to the graph of $y=\sqrt{x+1}$, we shift the graph to the left, and then for $y=2\cdot \sqrt{x+1}$, we need to stretch the graph by a factor $2$ away from the $x$-axis. (Alternatively, we could first stretch the the graph away from the $x$-axis, then shift the graph by $1$ to the left.)
$clipboard_ebf4193c52ac19fe58b16470a0e2d3e06.png$
For $y=-\left(\dfrac 1 x +2\right)$, we start with $y=\dfrac 1 x$ and add $2$, giving $y=\dfrac 1 x +2$, which shifts the graph up by $2$. Then, taking the negative gives $y=-\left(\dfrac 1 x +2\right)$, which corresponds to reflecting the graph about the $x$-axis. Note, that in this case, we cannot perform these transformations in the opposite order, since the negative of $y=\dfrac 1 x$ gives $y=-\dfrac 1 x$, and adding $2$ gives $y=-\dfrac 1 x+2$ which is not equal to $-(\dfrac 1 x+2)$.

$clipboard_e93e36a24fe48134375899d959d0df5b0.png$

We start with $y=x^3$. Adding $1$ in the argument, $y=(x+1)^3$, shifts its graph to the left by $1$. Then, taking the negative in the argument gives $y=(-x+1)^3$, which reflects the graph about the $y$-axis. Here, the order in which we perform these transformations is again important. In fact, if we first take the negative in the argument, we obtain $y=(-x)^3$. Then, adding one in the argument would give $y=(-(x+1))^3=(-x-1)^3$ which is different than our given function $y=(-x+1)^3$.

$clipboard_e331efdd9b34835cb9dd09585f3002ec4.png$

All these solutions may also easily be checked by using the graphing function of the calculator.

Example $\PageIndex{3}$

The graph of $f(x)=|x^3-5|$ is stretched away from the $y$-axis by a factor of $3$. What is the formula for the new function?
The graph of $f(x)=\sqrt{6x^2+3}$ is shifted up $5$ units, and then reflected about the $x$-axis. What is the formula for the new function?
How are the graphs of $y=2x^3+5x-9$ and $y=2(x-2)^3+5(x-2)-9$ related?
How are the graphs of $y=(x-2)^2$ and $y=(-x+3)^2$ related?

Solution

By Observation on page , we have to multiply the argument by $\frac 1 3$. The new function is therefore:
\[f\Big(\dfrac 1 3 \cdot x\Big)=\left|\Big(\dfrac 1 3\cdot x \Big)^3-5\right|=\left|\dfrac 1 {27} \cdot x^3-5\right| \nonumber\]
After the shift, we have the graph of a new function $y=\sqrt{6x^2+3}+5$.Then, a reflection about the $x$-axis gives the graph of the function $y=-(\sqrt{6x^2+3}+5)$.
By Observation on page , we see that we need to shift the graph of $y=2x^3+5x-9$ by $2$ units to the right.
The formulas can be transformed into each other as follows: \[\begin{align*}
\text{We begin with } & y=(x-2)^{2}\\
\text{Replacing x by x+5 gives } & y=((x+5)-2)^{2}=(x+3)^{2}\\
\text{Replacing x by -x gives } & y=((-x)+3)^{2}=(-x+3)^{2}
\end{align*} \nonumber \]Therefore, we have performed a shift to the left by $5$, and then a reflection about the $y$-axis.

We want to point out that there is a second solution for this problem: We begin with & $y=(x-2)^2$.
Replacing $x$ by $-x$ gives $y=((-x)-2)^2=(-x-2)^2$.
Replacing $x$ by $x-5$ gives $y=(-(x-5)-2)^2=(-x+5-2)^2=(-x+3)^2$. Therefore, we could also first perform a reflection about the $y$-axis, and then shift the graph to the right by $5$.

Some of the above functions have special symmetries, which we investigate now.

Definition: Even-Odd

A function $f$ is called even if $f(-x)=f(x)$ for all $x$.

Similarly, a function $f$ is called odd if $f(-x)=-f(x)$ for all $x$.

Example $\PageIndex{4}$

Determine, if the following functions are even, odd, or neither. $f(x)=x^2$, $g(x)=x^3$, & $h(x)=x^4$, $k(x)=x^5$,
$l(x)=4x^5+7x^3-2x$, & $m(x)=x^2+5x$.

Solution

The function $f(x)=x^2$ is even, since $f(-x)=(-x)^2=x^2$. Similarly, $g(x)=x^3$ is odd, $h(x)=x^4$ is even, and $k(x)=x^5$ is odd, since

\[\begin{aligned} g(-x)&=(-x)^3=-x^3=-g(x) \\ h(-x)&=(-x)^4=x^4=h(x) \\ k(-x)&=(-x)^5=-x^5=-k(x) \end{aligned}\]

Indeed, we see that a function $y=x^n$ is even, precisely when $n$ is even, and $y=x^n$ is odd, precisely when $n$ is odd. (These examples are in fact the motivation behind defining even and odd functions as in Definition Even-Odd above.)

Next, in order to determine if the function $l$ is even or odd, we calculate $l(-x)$ and compare it with $l(x)$.

\[\begin{aligned} l(-x)&=4(-x)^5+7(-x)^3-2(-x)\\ &=-4x^5-7x^3+2x \\ &=-(4x^5+7x^3-2x)\\ &=-l(x) \end{aligned}\]

Therefore, $l$ is an odd function.

Finally, for $m(x)=x^2+5x$, we calculate $m(-x)$ as follows:

\[m(-x)=(-x)^2+5(-x)=x^2-5x \nonumber \]

Note, that $m$ is not an even function, since $x^2-5x\neq x^2+5x$. Furthermore, $m$ is also not an odd function, since $x^2-5x\neq -(x^2+5x)$. Therefore, $m$ is a function that is neither even nor odd.

Observation: Even-Odd

An even function $f$ is symmetric with respect to the $y-$axis (if you reflect the graph of $f$ about the $y-$axis you get the same graph back), since even functions satisfy $f(-x)=f(x)$:

Example $y=x^2$:

$clipboard_ea45ba4599677e2a2bd473f698992b6c6.png$

An odd function $f$ is symmetric with respect to the origin (if you reflect the graph of $f$ about the $y-$axis and then about the $x-$axis you get the same graph back), since odd functions satisfy $f(-x)=-f(x)$:

Example $y=x^3$:

$clipboard_e822dc0a1fb7c4bdfb6b8062f41e8da9f.png$

Observation: Shift-Up-Down

Observation: Shift-Left-Right

Observation: Stretch

Observation: Stretch

Observation: Reflect

Example \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Definition: Even-Odd

Example \(\PageIndex{4}\)

Observation: Even-Odd