3.5 Transformation of functions  (Page 11/21)

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When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x -axis from a reflection with respect to the y -axis?

How can you determine whether a function is odd or even from the formula of the function?

For a function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ substitute $\text{\hspace{0.17em}}\left(-x\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}\left(x\right)\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ Simplify. If the resulting function is the same as the original function, $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right),\text{\hspace{0.17em}}$ then the function is even. If the resulting function is the opposite of the original function, $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right),\text{\hspace{0.17em}}$ then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

$\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is shifted up 1 unit and to the left 2 units.

$\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is shifted down 3 units and to the right 1 unit.

$g\left(x\right)=|x-1|-3$

$\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ is shifted down 4 units and to the right 3 units.

$\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ is shifted up 2 units and to the left 4 units.

$g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2$

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function $\text{\hspace{0.17em}}f.$

$y=f\left(x-49\right)$

$y=f\left(x+43\right)$

The graph of $\text{\hspace{0.17em}}f\left(x+43\right)\text{\hspace{0.17em}}$ is a horizontal shift to the left 43 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x+3\right)$

$y=f\left(x-4\right)$

The graph of $\text{\hspace{0.17em}}f\left(x-4\right)\text{\hspace{0.17em}}$ is a horizontal shift to the right 4 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x\right)+5$

$y=f\left(x\right)+8$

The graph of $\text{\hspace{0.17em}}f\left(x\right)+8\text{\hspace{0.17em}}$ is a vertical shift up 8 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x\right)-2$

$y=f\left(x\right)-7$

The graph of $\text{\hspace{0.17em}}f\left(x\right)-7\text{\hspace{0.17em}}$ is a vertical shift down 7 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x-2\right)+3$

$y=f\left(x+4\right)-1$

The graph of $f\left(x+4\right)-1$ is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of $f.$

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

$f\left(x\right)=4{\left(x+1\right)}^{2}-5$

$g\left(x\right)=5{\left(x+3\right)}^{2}-2$

decreasing on $\text{\hspace{0.17em}}\left(-\infty ,-3\right)\text{\hspace{0.17em}}$ and increasing on $\text{\hspace{0.17em}}\left(-3,\infty \right)$

$a\left(x\right)=\sqrt{-x+4}$

$k\left(x\right)=-3\sqrt{x}-1$

decreasing on $\left(0,\text{\hspace{0.17em}}\infty \right)$

Graphical

For the following exercises, use the graph of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ shown in [link] to sketch a graph of each transformation of $\text{\hspace{0.17em}}f\left(x\right).$

$g\left(x\right)={2}^{x}+1$

$h\left(x\right)={2}^{x}-3$

$w\left(x\right)={2}^{x-1}$

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

$f\left(t\right)={\left(t+1\right)}^{2}-3$

$h\left(x\right)=|x-1|+4$

$k\left(x\right)={\left(x-2\right)}^{3}-1$

$m\left(t\right)=3+\sqrt{t+2}$

Numeric

Tabular representations for the functions $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ are given below. Write $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ as transformations of $\text{\hspace{0.17em}}f\left(x\right).$

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −2 −1 −3 1 2
 $x$ −1 0 1 2 3 $g\left(x\right)$ −2 −1 −3 1 2
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −1 0 −2 2 3

$g\left(x\right)=f\left(x-1\right),\text{\hspace{0.17em}}h\left(x\right)=f\left(x\right)+1$

Tabular representations for the functions $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ are given below. Write $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ as transformations of $\text{\hspace{0.17em}}f\left(x\right).$

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −1 −3 4 2 1
 $x$ −3 −2 −1 0 1 $g\left(x\right)$ −1 −3 4 2 1
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −2 −4 3 1 0

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

$f\left(x\right)=|x-3|-2$

$f\left(x\right)=\sqrt{x+3}-1$

$f\left(x\right)={\left(x-2\right)}^{2}$

$f\left(x\right)=|x+3|-2$

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

$f\left(x\right)=-\sqrt{x}$

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

$f\left(x\right)=-{\left(x+1\right)}^{2}+2$

$f\left(x\right)=\sqrt{-x}+1$

For the following exercises, determine whether the function is odd, even, or neither.

$f\left(x\right)=3{x}^{4}$

even

$g\left(x\right)=\sqrt{x}$

$h\left(x\right)=\frac{1}{x}+3x$

odd

$f\left(x\right)={\left(x-2\right)}^{2}$

$g\left(x\right)=2{x}^{4}$

even

$h\left(x\right)=2x-{x}^{3}$

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function $\text{\hspace{0.17em}}f.$

$g\left(x\right)=-f\left(x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a vertical reflection (across the $\text{\hspace{0.17em}}x$ -axis) of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(-x\right)$

$g\left(x\right)=4f\left(x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a vertical stretch by a factor of 4 of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=6f\left(x\right)$

$g\left(x\right)=f\left(5x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal compression by a factor of $\text{\hspace{0.17em}}\frac{1}{5}\text{\hspace{0.17em}}$ of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(2x\right)$

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

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal stretch by a factor of 3 of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(\frac{1}{5}x\right)$

$g\left(x\right)=3f\left(-x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal reflection across the $\text{\hspace{0.17em}}y$ -axis and a vertical stretch by a factor of 3 of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=-f\left(3x\right)$

For the following exercises, write a formula for the function $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ that results when the graph of a given toolkit function is transformed as described.

The graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is reflected over the $\text{\hspace{0.17em}}y$ - axis and horizontally compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{4}$ .

$g\left(x\right)=|-4x|$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is reflected over the $\text{\hspace{0.17em}}x$ -axis and horizontally stretched by a factor of 2.

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ is vertically compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{3},\text{\hspace{0.17em}}$ then shifted to the left 2 units and down 3 units.

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

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

The graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is vertically compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{2},\text{\hspace{0.17em}}$ then shifted to the right 5 units and up 1 unit.

$g\left(x\right)=\frac{1}{2}{\left(x-5\right)}^{2}+1$

The graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

$g\left(x\right)=4{\left(x+1\right)}^{2}-5$

The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

$g\left(x\right)=5{\left(x+3\right)}^{2}-2$

$h\left(x\right)=-2|x-4|+3$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

$k\left(x\right)=-3\sqrt{x}-1$

$m\left(x\right)=\frac{1}{2}{x}^{3}$

The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ is compressed vertically by a factor of $\text{\hspace{0.17em}}\frac{1}{2}.$

$n\left(x\right)=\frac{1}{3}|x-2|$

$p\left(x\right)={\left(\frac{1}{3}x\right)}^{3}-3$

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

$q\left(x\right)={\left(\frac{1}{4}x\right)}^{3}+1$

$a\left(x\right)=\sqrt{-x+4}$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is shifted right 4 units and then reflected across the vertical line $\text{\hspace{0.17em}}x=4.$

For the following exercises, use the graph in [link] to sketch the given transformations.

$g\left(x\right)=f\left(x\right)-2$

$g\left(x\right)=-f\left(x\right)$

$g\left(x\right)=f\left(x+1\right)$

$g\left(x\right)=f\left(x-2\right)$

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