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This lesson will discuss how to apply different transformations to radical functions. It will also show how to identify these transformations in the graphs of different radical functions.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

The graph of the parent function $y=x $ and the graph of the radical function $y=ax−h +k$ are drawn on the same coordinate plane.

Find the values of $a,$ $h,$ and $k.${"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["2"]}}

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03148em;\">k<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["0"]}}

Explore

The graph of the function $y=x−h +k$ is shown in the coordinate plane. By changing the values of $h$ and $k,$ observe how the graph is horizontally and vertically translated.

Discussion

A translation of a function is a transformation that shifts a graph vertically or horizontally. As with the graph of any other function, a vertical translation of the graph of a radical function is achieved by adding some number to every output value of the function rule. Consider the parent function $y=x .$

$Functiony=x Vertical TranslationbykUnitsy=x +k $

If $k$ is a positive number, the translation is performed upwards. Conversely, if $k$ is negative, the translation is performed downwards. If $k=0,$ then there is no translation. This transformation can be shown on a coordinate plane.
A horizontal translation is instead achieved by subtracting a number from every input value.

$Functiony=x Horizontal TranslationbyhUnitsy=x−h $

In this case, if $h$ is a positive number, the translation is performed to the right. Conversely, if $h$ is negative, the translation is performed to the left. If $h=0,$ then there is no translation. This transformation can also be shown on a coordinate plane. Example

Ignacio has just started to learn about transformations of radical functions. In order to get some extra practice, he goes online to to find a worksheet for the topic. He found a pretty good exercise, which is divided into parts A, B, and C.
### Answer

### Hint

### Solution

Help Ignacio solve the exercise!

a **Equation:** $y=22x−6 −1$

**Graph:**

b **Equation:** $y=0.5x+2 +2$

**Graph:**

c **Equation:** $y=x+2 $

**Graph:**

a To translate the graph of $f$ to the right $3$ units, find an equation for $f(x−3).$

b To translate the graph of $g$ up $5$ units, find an equation for $g(x)+5.$

c To translate the graph of $h$ down $3$ units and to the left $4$ units, find an equation for $h(x+4)−3.$

$Functionf(x)=22x −1Translation3Units to the Righty=f(x−3)⇕y=22(x−3) −1 $

The right-hand side of the equation can be simplified by distributing the $2$ in the radicand.
Now both functions can be drawn on the same coordinate plane.
b To translate the graph of a function $5$ units up, the number $5$ must be added to the output of the function.

$Functiong(x)=0.5x+2 −3Translation5Units Upy=g(x)+5⇕y=0.5x+2 −3+5 $

The right-hand side of the equation can be simplified by adding the terms.
Now both functions can be drawn on the same coordinate plane.
c To translate the graph of a function $3$ units down and $4$ units to the left, $3$ must be subtracted from the output and $4$ must be added to the input.

$Functionh(x)=x−2 +3Translation3Units Downand4Units to the Lefty=h(x+4)−3⇕y=x+4−2 +3−3 $

The right-hand side of the equation can be simplified by subtracting the terms.
Now both functions can be drawn on the same coordinate plane.
Pop Quiz

The graphs of the rational function $y=x $ and a vertical or horizontal translation are shown in the coordinate plane.

Explore

The graph of the radical function $y=acx $ is shown in the coordinate plane. By changing the values of $a$ and $c,$ observe how the graph is vertically and horizontally stretched and shrunk.

Discussion

A function graph is vertically stretched or shrunk by multiplying the output of the function rule by some positive constant $a.$ Consider the radical function $y=x−1 +1.$
### Extra

Finding $a$ and $c$

$Functiony=x−1 +1Vertical Stretch/Shrink by a Factor ofay=a(x−1 +1)⇔y=ax−1 +a $

If $a$ is greater than $1,$ the graph is vertically stretched by a factor of $a.$ Conversely, if $a$ is less than $1,$ the graph is vertically shrunk by a factor of $a.$ If $a=1,$ then there is neither stretch nor shrink. All vertical distances from the graph to the $x-$axis are changed by the factor $a.$
Similarly, the graph of a radical function is horizontally stretched or shrunk by multiplying the input of the function rule by some positive constant $c.$ Consider this time the parent function $y=x .$

$Functiony=x Horizontal Stretch/Shrinkby a Factor ofcy=cx $

In this case, if $c$ is greater than $1,$ the graph is horizontally shrunk by a factor of $c.$ Conversely, if $c$ is less than $1,$ the graph is horizontally stretched by a factor of $c.$ If $c=1,$ then there is neither a stretch nor shrink of the graph.
Suppose that a function is horizontally or vertically stretched/shrunk, and that the graphs of the transformed and the original function are both drawn on the same coordinate plane. Then, the values of $a$ or $c$ can be found by following these procedures.

Finding $a$ | Select two points with the same $x-$coordinate, one point $P$ on the parent function and the other point $Q$ on the transformed function. The value of $a$ is the quotient of the $y-$coordinate of $Q$ and the $y-$coordinate of $P.$ |
---|---|

Finding $c$ | Select two points with the same $y-$coordinate, one point $P$ on the parent function and the other point $Q$ on the transformed function. The value of $c$ is the quotient of the $x-$coordinate of $P$ and the $x-$coordinate of $Q.$ |

Example

After mastering vertical and horizontal translations of radical functions, Ignacio is having a hard time understanding vertical and horizontal stretches and shrinks of this type of function. He asked for some help from his very good friend Jordan.

In order to help him, Jordan asks Ignacio to consider the following function.$f(x)=x−1 +3 $

She then tells him to complete the next two exercises. a Write the equation and draw the graph of a function whose graph is a vertical stretch of the graph of $f$ by a factor of $4.$

b Write the equation and draw the graph of a function whose graph is a horizontal shrink of the graph of $f$ by a factor of $2.$

a **Equation:** $y=4x−1 +12$

**Graph:**

b **Equation:** $y=2x−1 +3$

**Graph:**

a To vertically stretch the graph of $f$ by a factor of $4,$ the output of the function must be multiplied by $4.$

$Functionf(x)=x−1 +3Vertical Stretch by a Factor of4y=4f(x)⇕y=4(x−1 +3) $

The right-hand side of the function can be simplified by distributing the $4.$
Now both functions can be drawn on the same coordinate plane.
b To horizontally shrink the graph of $f$ by a factor of $2,$ the input of the function must be multiplied by $2.$

$Functionf(x)=x−1 +3Horizontal Shrink by a Factor of2y=f(2x)⇕y=2x−1 +3 $

Now both functions can be drawn on the same coordinate plane.
Pop Quiz

The graph of the parent function $y=x $ is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.

Discussion

A reflection of a function is a transformation that flips a graph over a line called the line of reflection. A reflection in the $x-$axis is achieved by changing the sign of every output value. This means changing the sign of the $y-$coordinate of every point on the graph of a function. Consider the radical function $y=x +1.$

$Functiony=x +1Reflection in thex-axisy=-(x +1)⇔y=-x −1 $

This reflection can be shown on a coordinate plane.
A reflection in the $y-$axis is instead achieved by changing the sign of every input value. In this case, note that the domain of the radical function needs to be modified from all non-negative numbers to all non-positive numbers. Consider this time the parent function $y=x .$

$Functiony=x Reflection in they-axisy=-x $

This transformation can also be shown on a coordinate plane. Example

Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.

Ignacio considers the following radical function.$f(x)=2x−2 −1 $

He wants to answer two practice exercises about reflection of radical functions. a Write the equation and draw the graph of a function whose graph is a reflection in the $x-$axis of the graph of $f.$

b Write the equation and draw the graph of a function whose graph is a reflection in the $y-$axis of the graph of $f.$

a **Equation:** $y=-2x−2 +1$

**Graph:**

b **Equation:** $y=2-x−2 −1$

**Graph:**

a To reflect the graph of the given function in the $x-$axis, the output needs to be multiplied by $-1.$

$Functionf(x)=2x−2 −1Reflection in thex-axisy=-f(x)⇕y=-(2x−2 −1) $

The right-hand side of the function can be simplified by distributing the $-1.$
Now both functions can be graphed on the same coordinate plane.
b To reflect the graph of the given function in the $y-$axis, the input needs to be multiplied by $-1.$

$Functionf(x)=2x−2 −1Reflection in they-axisy=f(-x)⇕y=2-x−2 −1 $

Now, both functions can be graphed on the same coordinate plane.
Example

Ignacio confidently states that he can now solve any exercise about transformations of radical functions. Jordan skeptically challenges her friend to solve an exercise that combines transformations.

Help Ignacio solve Jordan's challenge!**Equation:** $y=-x+5 −6$

**Graph:**

Start by multiplying the output by $2$ to find the function that represents the vertical stretch. Then, multiply the input by $-1$ to reflect the graph in the $y-$axis. Finally, subtract $3$ units from the input.

To help Ignacio, the transformations will be applied one at a time.

$Functionf(x)=0.5x+2 −3Vertical Stretch by a Factor of2y=2f(x)⇕y=2(0.5x+2 −3) $

Simplify the right-hand side of the above equation by distributing the $2.$
$y=2(0.5x+2 −3)$

Distr

Distribute $2$

$y=1x+2 −6$

IdPropMult

Identity Property of Multiplication

$y=x+2 −6$

$g(x)=x+2 −6 $

$Functiong(x)=x+2 −6Reflection in they-axisy=g(-x)⇕y=-x+2 −6 $

The graph of this function is a reflection in the $y-$axis of the graph of $g.$ Let $h(x)$ be this function.
$h(x)=-x+2 −6 $

$Functionh(x)=-x+2 −6Translation3Units to the Righty=h(x−3)⇕y=-(x−3)+2 −6 $

Simplify the right-hand side of the equation by distributing the $-1$ and adding the terms in the radicand.
The graph of this final function is a vertical stretch by a factor of $2,$ followed by a reflection in the $y-$axis, and a translation $3$ units to the right of the graph of $f.$ Closure

With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graphs of the radical function $y=ax−h +k$ and its corresponding parent function $y=x $ are given.

Find the values of $a,$ $h,$ and $k.$Start by considering a vertical stretch. Then, consider vertical and horizontal translations.

Disregard translations for a moment. In the graph of $y=x $ it can be understood as, after moving $1$ unit to the right of the starting point $(0,0),$ the graph increases vertically $1$ unit. Conversely, in the graph of $y=ax−h +k,$ after moving $1$ unit to the right of the starting point $(-3,0),$ the graph increased vertically $2$ units.

$Functiony=x Vertical Stretch bya Factor of2y=2x $

Next, pay close attention to the starting point of each curve. It can be concluded that the graph of the parent function $y=x $ needs to be translated $3$ units to the left to match the other graph. Therefore, the value of $h$ is $-3.$
$Functiony=2x Horizontal Translation3Units to the Lefty=2x−(-3) $

Finally, note that the $y-$coordinate of the initial point of both graphs is $0.$ Therefore, there is no vertical translation. This means that $k=0.$
$Functiony=2x−(-3) Vertical Translation0Units Upy=2x−(-3) +0 $

It has been found that $a=2,$ $h=-3,$ and $k=0.$ The obtained function can be simplified.
The transformations can be shown on a graph.