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Translating Radical Functions

The equation of the standard form of a radical function is y equals the square root of x

To translate the graph of the function, you will study the vertex form of a radical function which is y equals a. the square root of x minus h plus k

In this form, (h, k) represents the vertex of the function and will tell you how the graph is translated. The value of a will tell you how the graph opens.

Notice the vertex of the function is (h, k). The expression under the radical is x - h, so you will determine the opposite of the h when stating the vertex.

y equals the square root of x minus 3 plus 4

In the function above, the vertex would be (3, 4) because the value of h is the opposite of −3 which is 3 and the value of k is 4.

Examples #1 and #2

Watch Find the Vertex of a Radical Function Given Its Equation.

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Translating the Parent Graph

The vertex will tell you how the graph of the function is translated from the parent graph.

How is the function below translated from y equals the square root of x

y equals the square root of x plus 7 minus 1

The vertex for this function is at (−7, −1). This means the graph is translated seven units left and one unit down.

Examples #3 and #4

Watch Describe the Translation of a Radical Function from Its Parent Graph Given Its Equation.

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Example #5

Watch Describe the Translation of a Radical Function from Its Parent Graph Given Its Graph.

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The importance of a

When referring back to the vertex form of a radical function, the value of a will determine whether your graph will open up or down. The value of a also effects the steepness of the graph.

Open Up or Down?

Given the equation y equals a. the square root of x minus h plus k , you can easily determine if the graph will open up or down. Consider the two cases below.

The parent graph of the square root function is graphed on a coordinate plane with a vertex at (0,0) and passes through the following points: (1, 1) and (4,2) . The graph goes up and to the right.
a > 0
The parent graph of the square root function reflected over the x-axis is graphed on a coordinate plane with a vertex at (0,0) and passes through the following points: (1, negative 1) and (4, negative 2) . The graph goes down and to the right.
a < 0

Examples #6 and #7

Watch Describe Key Features of a Radical Function Given Its Equation.

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Stretching and Shrinking

You can easily identify stretching and shrinking on a graph. The red and purple graphs below have been stretched and the blue graph was shrunk. Notice the values of a.

A coordinate plane with 4 square root functions graphed all having a vertex at (0,0) and going up and to the right. The closest graph to the y-axis is y = 3 times the square root of x. The next closest to the y-axis is y = 2 times the square root of x. The next closest to the y-axis is y = the square root of x. The graph that is the farthest from the y-axis is y = 0.5 times the square root of x.

Example #8 Graphing

Watch Graph a Radical Function and Describe Its Translation from the Parent Graph.

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Example #9

Watch Graph a Radical Function and Describe Its Translation from the Parent Graph.

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