Learn – Algebra foundations – Function transformations

1 Motivation
2 Example transformations
- 2.1 Exercises
  - 2.1.1 Problem set
  - 2.1.2 Problem set
3 Shift in Y-direction
- 3.1 Exercises
4 Reflection in Y-direction
- 4.1 Exercises
5 Stretch in Y-direction
- 5.1 Exercises
6 Shift in X-direction
- 6.1 Exercises
7 Reflection in X-direction
- 7.1 Exercises
8 Stretch in X-direction
- 8.1 Exercises
9 Combinations
- 9.1 Exercises

Motivation

Say, we are asked to draw the graph of $f(x) = x^2$ . How would we do this? We would like to find some example points that satisfy $f(x) = x^2$ . To do that, let us make a table and find some $x-y$ pairs.

$\boldsymbol{x}$	$\boldsymbol{y}$
$-3$	$9$
$-2$	$4$
$-1$	$1$
$0$	$0$
$1$	$1$
$2$	$4$
$3$	$9$

We plot example points for $x-y$ pairs from the table and since we assume the domain of the function to be $\mathbb{R}$ , we connect the example points to form a continuous curve.

All good so far.

Say, now we are asked to draw the graph of $g(x) = x^2+1$ . We could of course follow the same method as we did for the previous example. But, don’t you think it is too much work to do? What if we are asked next to graph $h(x) = x^2-2$ or $l(x) = -2x^2$ ?

What we can observe is that the functions $x^2+1, x^2-2$ and $-2x^2$ seem to be closely related to the function $x^2$ that we have already taken the trouble to draw. Can we somehow use the graph of $x^2$ and just modify it slightly as required for the related graphs? That would not only be a lot quicker and less laborious, but also has less possibility of making errors.

So, how do we understand the relations between a known function such as $x^2$ and a new function such as $x^2+1$ or $-2x^2$ ? And, how do we use this understanding to draw the graphs of the new functions?

This is the subject of the very important topic of function transformations.

Example transformations

Exercises

Problem set

In the following, assume $f(x)$ is the parent function and $g(x)$ is the derived function. The explicit expression for $f(x)$ is given, and $g(x)$ is given as a function transformation of $f(x)$ . Find the explicit expression for $g(x)$ .

$f(x) = x^5,\,\, g(x) = f(x)+2$
$f(x) = \frac{1}{x},\,\, g(x) = f(x)-2$
$f(x) = x^2,\,\, g(x) = 2f(x)$
$f(x) = 7^x,\,\, g(x) = -f(x)$
$f(x) = \sqrt{x},\,\, g(x) = -2f(x)$
$f(x) = \frac{1}{x},\,\, g(x) = -7f(x)+9$

Problem set

In the following, assume $f(x)$ is the parent function and $g(x)$ is the derived function. The explicit expressions for $f(x)$ and $g(x)$ are given. Express $g(x)$ as a function transformation of $f(x)$ .

$f(x) = \sqrt{x},\,\, g(x) = 3+\sqrt{x}$
$f(x) = 3^x,\,\, g(x) = 7-3^x$
$f(x) = x^{10},\,\, g(x) = 2x^{10}$
$f(x) = \frac{1}{x^2},\,\, g(x) = \frac{5}{x^2}$
$f(x) = 5^x,\,\, g(x) = -3\times 5^x-4$

Shift in Y-direction

Exercises

Problem set

In each of the following problems, a function $f(x)$ is assumed. A new function $g(x)$ is defined as a function transformation of $f(x)$ . And, the graph shown is the graph of $f(x)$ . Draw the graph of $g(x)$ .

$g(x) = f(x)+3$

$g(x) = f(x)-1$

$g(x) = f(x)-2$

$g(x) = f(x)+0.5$

Problem set

In each of the following problems, a parent function $f(x)$ is given, and a derived function $g(x)$ is given as a function transformation of $f(x)$ . Give the explicit expression for $g(x)$ . Also, explain how the graph of $g(x)$ is related to the graph of $f(x)$ .

$f(x) = x^3, g(x) = f(x)+2$
$f(x) = \sqrt{x}, g(x) = f(x)-2$
$f(x) = 2^x, g(x) = f(x)+1.5$
$f(x) = \frac{1}{x}, g(x) = f(x) - 10$
$f(x) = -x^2+x^3, g(x) = f(x) + 3.9$
$f(x) = -\sqrt{x}-x^2, g(x) = f(x) - \pi$

Problem set

In each of the following problems, explicit functions for $f(x)$ and $g(x)$ are given. Express $g(x)$ as a function transformation of $f(x)$ . Also, explain how the graph of $g(x)$ is related to the graph of $f(x)$ .

$f(x) = x^2, g(x) = x^2+5$
$f(x) = 11^x, g(x) = 11^x-2$
$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{x} + \frac{1}{3}$
$f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}, g(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} - 3.3$
$f(x) = \log x, g(x) = \log x - 1.5$

Reflection in Y-direction

Exercises

Problem set

$g(x) = -f(x)$

$g(x) = -f(x)$

$g(x) = -f(x)$

$g(x) = -f(x)$

Problem set

$f(x) = x^3, g(x) = -f(x)$
$f(x) = \sqrt{x}, g(x) = -f(x)$
$f(x) = 2^x, g(x) = -f(x)$
$f(x) = \frac{1}{x}, g(x) = -f(x)$
$f(x) = x^2+x^3, g(x) = -f(x)$

Problem set

$f(x) = x^2, g(x) = -x^2$
$f(x) = \sqrt[3]{x}, g(x) = -\sqrt[3]{x}$
$f(x) = 11^x, g(x) = -11^x$
$f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}, g(x) = -\frac{1}{x} - \frac{1}{x^2} - \frac{1}{x^3}$
$f(x) = \frac{x^2+1}{x+2}, g(x) = \frac{-x^2-1}{x+2}$
$f(x) = \sin x, g(x) = -\sin x$

Stretch in Y-direction

Exercises

Problem set

$g(x) = 2f(x)$

$g(x) = 3f(x)$

$g(x) = \frac{1}{2}f(x)$

$g(x) = \frac{f(x)}{3}$

Problem set

$f(x) = x^3, g(x) = 2f(x)$
$f(x) = \sqrt{x}, g(x) = \frac{f(x)}{2}$
$f(x) = 2^x, g(x) = 1.5f(x)$
$f(x) = \frac{1}{7^x}, g(x) = 0.5f(x)$
$f(x) = \frac{3}{x+1}, g(x) = 3.9f(x)$
$f(x) = \frac{x+1}{x^3-2x^2-1}, g(x) = \frac{f(x)}{5}$
$f(x) = \sin x, g(x) = \frac{1}{10}f(x)$

Problem set

$f(x) = x^2, g(x) = 5x^2$
$f(x) = \sqrt[3]{x}, g(x) = \frac{1}{3}\sqrt[3]{x}$
$f(x) = 11^x, g(x) = \frac{11^x}{2}$
$f(x) = \frac{1}{5^x}, g(x) = \frac{3}{5^x}$
$f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}, g(x) = -\frac{3.3}{x} - \frac{3.3}{x^2} - \frac{3.3}{x^3}$
$f(x) = \frac{1}{x}, g(x) = \frac{1}{2x}$
$f(x) = \log x + e^x, g(x) = 2\log x + 2e^x$

Shift in X-direction

Exercises

Problem set

$g(x) = f(x+1)$

$g(x) = f(x-10)$

$g(x) = f(x+2)$

$g(x) = f(x-2)$

Problem set

$f(x) = x^3, g(x) = f(x+1)$
$f(x) = \sqrt{x}, g(x) = f(x-2)$
$f(x) = 2^x, g(x) = f(x+1.5)$
$f(x) = \frac{1}{x}, g(x) = f(x-10)$
$f(x) = \frac{1}{7^x}, g(x) = f(x+3)$
$f(x) = x^2+x^3, g(x) = f(x+3.9)$
$f(x) = \sqrt{x}-x^2, g(x) = f(x-99)$

Problem set

$f(x) = x^2, g(x) = (x+5)^2$
$f(x) = x^5, g(x) = (x-1)^5$
$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{x+ \frac{1}{3}}$
$f(x) = 11^x, g(x) = 11^{x-2}$
$f(x) = \frac{1}{x}, g(x) = \frac{1}{x-4}$
$f(x) = \frac{1}{x^2}, g(x) = \frac{1}{(x+2)^2}$
$f(x) = x^3 + x^2, g(x) = (x+1)^3 + (x+1)^2$
$f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}, g(x) = \frac{1}{x-2.5} + \frac{1}{(x-2.5)^2} + \frac{1}{(x-2.5)^3}$
$f(x) = \log x, g(x) = \log (x - 1.5)$

Reflection in X-direction

Exercises

Problem set

$g(x) = f(-x)$

$g(x) = f(-x)$

$g(x) = f(-x)$

$g(x) = f(-x)$

Problem set

$f(x) = 0.5^x, g(x) = f(-x)$
$f(x) = x^3+3, g(x) = f(-x)$
$f(x) = \frac{1}{7^x}, g(x) = f(-x)$
$f(x) = \sqrt[5]{x}, g(x) = f(-x)$
$f(x) = x^2, g(x) = f(-x)$
$f(x) = \frac{1}{x}, g(x) = f\left(-\frac{1}{10}(x+10)\right)$
$f(x) = x^2+x^3, g(x) = f\left(-3.9(x-3.9)\right)$

Problem set

$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{-x}$
$f(x) = 11^x, g(x) = 11^{-x}$
$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{-\frac{1}{3}x}$
$f(x) = \log x, g(x) = \log \left(-\frac{x}{1.5}\right)$
$f(x) = \sin x, g(x) = \sin \left(-2(x+\pi)\right)$

Stretch in X-direction

Exercises

Problem set

$g(x) = f(5x)$

$g(x) = f(3x)$

$g(x) = f\left(\frac{1}{2}x\right)$

$g(x) = f\left(\frac{x}{3}\right)$

Problem set

$f(x) = x^3, g(x) = f(2x)$
$f(x) = \sqrt{x}, g(x) = f\left(\frac{x}{2}\right)$
$f(x) = 2^x, g(x) = f(1.5x)$
$f(x) = \sin x, g(x) = f\left(\frac{x}{10}\right)$
$f(x) = \frac{1}{x}, g(x) = f(3.9x)$
$f(x) = \frac{x+1}{x^3-2x^2-1}, g(x) = f\left(\frac{1}{\pi}x\right)$

Problem set

$f(x) = x^2, g(x) = (5x)^2$
$f(x) = 11^x, g(x) = 11^{x/2}$
$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{\frac{1}{3}x}$
$f(x) = \log x, g(x) = \log \left(\frac{x}{1.5}\right)$
$f(x) = x^3 + x^2, g(x) = (1.3x)^3 + (1.3x)^2$
$f(x) = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}, g(x) = \frac{1}{0.2x} + \frac{1}{(0.2x)^2} + \frac{1}{(0.2x)^3}$

Combinations

Exercises

Problem set

In each of the following problems, a function $f(x)$ is assumed. A new function $g(x)$ is defined in terms of $f(x)$ . And, the graph shown is the graph of $f(x)$ . Form a sequence of intermediate derived functions ( $p(x), q(x),\cdots$ ), draw the graph of each intermediate derived function, and finally draw the graph of $g(x)$ .

$g(x) = -f(x)+2$

$g(x) = 2f(x)+1$

$g(x) = \frac{1}{2}f(x)-1$

$g(x) = -2f(x+2)$

$g(x) = 2f(x+2)+2$

$g(x) = 2f(x-1.55)+1$

$g(x) = -\frac{1}{2}f(x-2)+4$

Problem set

In each of the following problems, a function $f(x)$ is assumed. A new function $g(x)$ is defined in terms of $f(x)$ . And, the graph shown is the graph of $f(x)$ . Draw the graph of $g(x)$ .

$g(x) = -f(x-3)$

$g(x) = 2f(3x)-1$

$g(x) = f(5(x-5))$

$g(x) = f(5x-5)$

$g(x) = 2f(2(x-2))-1$

$g(x) = 2f(-3(x-3))-1$

$g(x) = -2f\left(-\frac{1}{2}(x+3)\right)-1$

Problem set

For each of the following problems, give the explicit expression for $g(x)$ .

$f(x) = x^3,\,\,p(x) = 2f(x),\,\,g(x) = p(x)+1$
$f(x) = 2^x,\,\,p(x) = -f(x),\,\,q(x) = 2p(x),\,\,g(x) = q(x)-2$
$f(x) = \frac{1}{x},\,\,p(x) = -f(x),\,\,q(x) = \frac{p(x)}{2},\,\,g(x) = q(x)+3$
$f(x) = x^2,\,\,p(x) = f(x-2),\,\,q(x) = -p(x),\,\,g(x) = q(x)+2$
$f(x) = \frac{1}{x^3},\,\,p(x) = f(x+1),\,\,q(x) = \frac{p(x)}{3},\,\,r(x) = -q(x),\,\,g(x) = r(x) - 1$

Problem set

In each of the following problems, a parent function $f(x)$ is given, and a derived function $g(x)$ is given in terms of the $f(x)$ . Give the explicit expression for $g(x)$ . Also, giving the sequence of intermediate functions( $p(x), q(x),\cdots$ ) to transform $f(x)$ into $g(x)$ , explain how the graph of $g(x)$ is related to the graph of $f(x)$ .

$f(x) = x^3, g(x) = 2f(x+2)+2$
$f(x) = x^2, g(x) = 5f(x+5)+5$
$f(x) = 2^x, g(x) = -0.5f(x+1)-2$
$f(x) = x^2, g(x) = -\frac{1}{3}f(x-3)-3$
$f(x) = \frac{1}{x}, g(x) = -\frac{f(x-7)}{3}+7$

Problem set

In each of the following problems, $f(x)$ and $g(x)$ are defined. Express $g(x)$ is terms of $f(x)$ . Also, giving the sequence of intermediate functions( $p(x), q(x),\cdots$ ) to transform $f(x)$ into $g(x)$ , explain how the graph of $g(x)$ is related to the graph of $f(x)$ .

$f(x) = x^3, g(x) = 2(x+2)^3+2$
$f(x) = x^2, g(x) = \frac{1}{5}(x+5)^2+5$
$f(x) = 2^x, g(x) = 0.1\times2^{x+1}-3$
$f(x) = \frac{1}{x}, g(x) = \frac{1}{2x}+1$
$f(x) = x^5, g(x) = -\frac{(x-3)^2}{3}-3$

Problem set

$f(x) = \tan x, g(x) = -3f\left(\pi\left(x+2\right)\right)$
$f(x) = \sin x, g(x) = 2f \left(-3\left(x-\frac{\pi}{2}\right)\right)-1$
$f(x) = \sqrt{x}, g(x) = -2f(x-3) + 2$

Problem set

$f(x) = \tan x, g(x) = -3\tan\left(\pi\left(x+2\right)\right)$
$f(x) = \sin x, g(x) = 2\sin \left(-3\left(x-\frac{\pi}{2}\right)\right)-1$
$f(x) = \sqrt{x}, g(x) = -2\sqrt{x-3} + 2$

Problem set

For each of the following problems, first plot $f(x)$ . Then, using your knowledge of transformations (translation, reflection, scaling) plot $g(x)$ .

$f(x) = |x|, g(x) = \frac{1}{2}|x|$
$f(x) = \frac{1}{2}|x|, g(x) = -\frac{1}{2}|x|+3$
$f(x) = \sqrt[3]{x}, g(x) = \sqrt[3]{x}-10$
$f(x) = \sin x, g(x) = 3\sin x + 3$
$f(x) = \sec x, g(x) = \sec x - 1$

Problem set

For problems 1-5, first plot $f(x)$ . Then, using your knowledge of transformations (translation, reflection, scaling) plot $g(x)$ .

$f(x) = \sqrt{x}, g(x) = \sqrt{x-4}$
$f(x) = x^2, g(x) = 3x^2-3$
$f(x) = \sqrt{x}, g(x) = \sqrt{4x+4}$
$f(x) = \sin x, g(x) = \sin\left(2\pi x\right)$
$f(x) = \cos x, g(x) = 2\cos\left(2\pi x-\frac{\pi}{2}\right)$
Use algebra to justify that the graph of $\sin x$ is same as graph of $\cos x$ shifted right by $\frac{\pi}{2}$ . Hint: $\sin x = \cos\left(\frac{\pi}{2}-x\right)=\cos\left(x - \frac{\pi}{2}\right)$
Use algebra to justify that the graph of $\cot x$ is same as graph of $\tan x$ shifted right by $\frac{\pi}{2}$ and then reflected about $Y$ -axis.

Problem set

For each of the following problems, give the sequence of transformations (translation, reflection, scaling) needed to be done on function $f$ to obtain function $g$ .

$f(x) = x^2, g(x) = x^2 - 3$
$f(x) = 3^{1.5x}, g(x) = \frac{3^{1.5x}}{2}$
$f(x) = \ln x, g(x) = \ln (x-2)$
$f(x) = \ln x, g(x) = \ln \frac{x}{2}$
$f(x) = e^{2x}, g(x) = -e^{2x}$
$f(x) = e^{2x}, g(x) = e^{-2x}$

Problem set

In the following problems, from your knowledge on graphs of $f(x)$ , plot $g(x)$ . (Assume appropriate domain for the functions, and assume $\sqrt{x}$ refers to the positive square root of $x$ )

$f(x) = 2^x, g(x) = -2^{-x}$
$f(x) = x^2, g(x) = -(x+3)^2 - 3$
$f(x) = \log x, g(x) = \log (x + 1)$
$f(x) = \ln x, g(x) = -\ln (x - 5)$
$f(x) = \sqrt{x}, g(x) = -\sqrt{x-5}$

Problem set

For each of the following problems, give the sequence of transformations (translation, reflection, scaling) needed to be done on function $f$ to obtain function $g$ . (Assume $\sqrt{x}$ refers to the positive square root of $x$ )

$f(x) = x^4 - 5, g(x) = \frac{x^4}{16} - 1$
$f(x) = x^3 + 8, g(x) = -2(x+2)^3$
$f(x) = 2^x, g(x) = \frac{2^x}{2^{3x}}-2$
$f(x) = \sqrt{x} + 5, g(x) = \sqrt{25x+25}$
$f(x) = 3\sqrt{3^x}, g(x) = \sqrt{9^x9^3}$

Problem set

$f(x) = 2^x, g(x) = (\sqrt{2})^x(\sqrt{2})^x(\sqrt{2})^x(\sqrt{2})^x$
$f(x) = 16^x, g(x) = -4^x$
$f(x) = \sqrt{3^x + 3}, g(x) = \sqrt{3^{x+2}+27}$
$f(x) = 25^x, g(x) = \frac{5^{-2x}}{5} + 5$
$f(x) = 5^x, g(x) = \frac{5^{-\frac{x}{2}} + 3}{-0.5}$

Problem set

In the following problems, from your knowledge on graphs of $f(x)$ , plot $g(x)$ .

$f(x) = 3^x, g(x) = -\frac{3^x}{3^{3x}}$
$f(x) = \log x, g(x) = \log x + \log 2$
$f(x) = \log x, g(x) = \frac{\log x^3}{6}$