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Describing Transformations of Radical Functions

The common transformations can be applied to radical functions as usual.

Rule

Translation

By adding some number to every function value, g(x)=f(x)+k, g(x) = f(x) + k, a function's graph is translated vertically.

Translate graph upward

A graph is translated horizontally by subtracting a number from the input of the function rule. g(x)=f(xh) g(x) = f(x - h) Note that the number, h,h, is subtracted and not added. This is so that a positive hh leads to a translation to the right, which is the positive xx-direction.

Translate graph to the right

Rule

Reflection

A function is reflected in the xx-axis by changing the sign of all function values: g(x)=-f(x). g(x) = \text{-} f(x). Graphically, all points on the graph move to the opposite side of the xx-axis, while maintaining their distance to the xx-axis.

Reflect graph in xx-axis

A graph is instead reflected in the yy-axis by moving all points on the graph to the opposite side of the yy-axis. This occurs by changing the sign of the input of the function. g(x)=f(-x) g(x) = f(\text{-} x) Notice that the yy-intercept is preserved.

Reflect graph in yy-axis

Rule

Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant a>0a > 0: g(x)=af(x). g(x) = a \cdot f(x). All vertical distances from the graph to the xx-axis are changed by the factor a.a. Thus, preserving any xx-intercepts.

Stretch graph vertically

By instead multiplying the input of a function rule by some constant a>0,a > 0, g(x)=f(ax), g(x) = f(a \cdot x), its graph will be horizontally stretched or shrunk by the factor 1a.\frac 1 a. Since the xx-value of yy-intercepts is 0,0, they are not affected by this transformation.

Stretch graph horizontally

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Exercise

The graphs of four radical functions are shown in the image.

Compare the graphs with each other or their parent function to match each with its corresponding function rule: f(x)=-x+1+0.3g(x)=-x+10.3h(x)=x+10.3t(x)=x+130.3.\begin{aligned} f(x) &= \text{-} \sqrt{x + 1} + 0.3 & g(x) &= \sqrt{\text{-} x + 1} - 0.3\\ h(x) &= \sqrt{x + 1} - 0.3 & t(x) &= \sqrt[3]{x + 1} - 0.3. \end{aligned}

Show Solution
Solution

The radicals in the function rules are either square roots or cube roots. Thus, the parent function of each is either the square root of x,x, or the cube root of x.x. Let's start with Graph I.

Example

Graph I

From the image, we can see that the function corresponding to Graph I isn't defined for all real numbers. Thus, it must be one of the square root functions. Comparing it to the graph of k(x)=x,k(x) = \sqrt{x}, can give us more information about its function rule.

It looks as though Graph I is a translation of k(x)=x,k(x) = \sqrt{x}, both to the left and downward. Thus, its function rule is similar to k(x),k(x), but with some number added to the input and some number subtracted from the output. Among the available choices, there is a match, h(x).h(x). I:  h(x)=x+10.3 \text{I:}\ \ h(x) = \sqrt{x + 1} - 0.3

Example

Graph II

Graph II is very similar to Graph I, so let's compare them.

Here, II looks to be a reflection of I in the xx-axis. If this is the case, one of the options must be equal to -h(x).\text{-} h(x).

-h(x)\text{-} h(x)
-(x+10.3)\text{-} \left( {\color{#0000FF}{\sqrt{x + 1} - 0.3}} \right)
-x+1+0.3\text{-} \sqrt{x + 1} + 0.3

We can now identify that the function f(x)=-x+1+0.3 f(x) = \text{-} \sqrt{x + 1} + 0.3 is a reflection of hh in the xx-axis. Thus, it corresponds to Graph II.

Example

Graph III

Graph III has to be the graph of one of the square root functions, as it's not defined for all real numbers. There is only one square root function remaining, g,g, which must be its match. III:  g(x)=-x+10.3. \text{III:}\ \ g(x) = \sqrt{\text{-} x + 1} - 0.3. Looking at the graph, we can confirm that this is the case. Comparing g(x)g(x) with h(x),h(x), the sign of the input has been reversed. Thus gg is a reflection of hh in the yy-axis. This is exactly what can be seen in the graphs when comparing I and III.

Example

Graph IV

With only one rule remaining, Graph IV must correspond to t.t. To confirm this, tt can be viewed as a translation of l(x)=x3,l(x) = \sqrt[3]{x}, 11 unit to the left and 0.30.3 units downward. This can also be seen in its graph.

Thus, we have matched all graphs and function rules. I: h(x)II: f(x)III: g(x)IV: t(x)\begin{aligned} \text{I: }&h(x) & \text{II: }&f(x)\\ \text{III: }&g(x) & \text{IV: }&t(x) \end{aligned}

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Exercise


The rules of ff and gg are given such that gg is a transformation of f.f. f(x)=2x1g(x)=-f(x+1) f(x) = \sqrt{2x} - 1 \qquad g(x) = \text{-} f(x + 1) Express gg as a function of x.x. Then, graph both functions in the same coordinate plane and state the transformation(s) ff underwent to become g.g.

Show Solution
Solution
To write gg as a function of x,x, we can first find an expression for f(x+1),f(x+1), then one for -f(x+1).\text{-} f(x+1). This is done by replacing xx with x+1x+1 in the function rule of f.f. f(x+1)=2(x+1)1 f({\color{#0000FF}{x+1}})=\sqrt{2({\color{#0000FF}{x+1}})}-1 Now that we have an expression for f(x+1),f(x+1), we can find -f(x+1),\text{-} f(x+1), by multiplying the expression above by -1.\text{-} 1. This gives the rule for g.g.
g(x)=-f(x+1)g(x) = \text{-} f(x + 1)
g(x)=-(2(x+1)1)g(x) = \text{-}\left({\color{#0000FF}{\sqrt{2(x+1)}-1}}\right)
g(x)=-2(x+1)+1g(x) = \text{-}\sqrt{2(x+1)}+1
g(x)=-2x+2+1g(x)=\text{-}\sqrt{2x+2}+1
The function gg is now written in terms of xx without containing the function f.f. We'll graph ff and g.g.
Example

Graphing f(x)=2x1f(x)=\sqrt{2x}-1

To graph f,f, let's calculate some function values in a table. Since ff is a square root function, it's not defined when the expression under the radical sign is negative. Thus, a good starting value is x=0.x=0.

xx 2x1\sqrt{2x}-1 f(x)f(x)
0{\color{#0000FF}{0}} 2(0)1\sqrt{2({\color{#0000FF}{0}})}-1 -1\text{-}1
1{\color{#0000FF}{1}} 2(1)1\sqrt{2({\color{#0000FF}{1}})}-1 0.41\sim 0.41
2{\color{#0000FF}{2}} 2(2)1\sqrt{2({\color{#0000FF}{2}})}-1 11
3{\color{#0000FF}{3}} 2(3)1\sqrt{2({\color{#0000FF}{3}})}-1 1.45\sim 1.45
4{\color{#0000FF}{4}} 2(4)1\sqrt{2({\color{#0000FF}{4}})}-1 1.83\sim 1.83

The xx-values and the function values can now be plotted as points (x,f(x))(x,f(x)) in a coordinate plane. We'll connect the points with a smooth curve. Note that the function is not defined for x<0.x<0.

Example

Graphing g(x)=-2x+2+1g(x) = \text{-}\sqrt{2x + 2}+1

We'll use the same procedure as we did for graphing f.f. The function gg is also a square root function and is not defined for xx-values less than -1.\text{-}1. Therefore, we will start at x=-1x=\text{-}1 for our table values.

xx -2x+2+1\text{-}\sqrt{2x+2}+1 f(x)f(x)
-1{\color{#0000FF}{\text{-}1}} -2(-1)+2+1\text{-}\sqrt{2({\color{#0000FF}{\text{-}1}})+2}+1 11
0{\color{#0000FF}{0}} -2(0)+2+1\text{-}\sqrt{2({\color{#0000FF}{0}})+2}+1 -0.41\sim\text{-}0.41
1{\color{#0000FF}{1}} -2(1)+2+1\text{-}\sqrt{2({\color{#0000FF}{1}})+2}+1 -1\text{-}1
2{\color{#0000FF}{2}} -2(2)+2+1\text{-}\sqrt{2({\color{#0000FF}{2}})+2}+1 -1.45\sim\text{-}1.45
3{\color{#0000FF}{3}} -2(3)+2+1\text{-}\sqrt{2({\color{#0000FF}{3}})+2}+1 -1.83\sim\text{-}1.83

The points can now be plotted in the same coordinate plane as f,f, again connecting the points with a smooth curve.

Example

Transformations from ff to gg

To determine the transformations ff underwent to become g,g, we'll study the function rule of g:g\text{:} g(x)=-f(x+1). g(x) = {\color{#FF0000}{\text{-}}} f(x{\color{#0000FF}{+1}}). This function can be seen as adding 1{\color{#0000FF}{1}} to the input of f,f, and multiplying the function value by -1.{\color{#FF0000}{\text{-} 1}}. Therefore, the first transformation is a translation 11 unit to the left.

Furthermore, the second transformation is a reflection in the xx-axis.

Therefore, ff has undergone both a horizontal translation and a reflection in the xx-axis to become g.g.

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