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Wize High School Algebra I Textbook (Common Core) > Radical Equations and Functions

Graphs of Radical Functions

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Graphs of Radical Functions

Basic Shape

The basic shape for the graph depends on the index of the radical. This may look like an "s" type shape, or a gradual curve.


The index plays a large part when determining the overall shape of the graph
  • Odd index - domain is all real numbers. Graph is "s" shaped over all real numbers.
  • Even index - domain is restricted so that only non-negative numbers end up under the radical. Has a curve for only some values.
Example 1
Graph the following radical functions using key values.

f(x)=x4f(x) = \sqrt[4]{x}

ANSWER:

Shifted graphs

Once we have the basic shape, we can look at slightly more complicated graphs. These are simply shifted by a given amount.
  • Adding to the inside will shift the basic graph left
  • Adding to the outside will shift the basic graph up
  • Subtracting is the same as adding a negative number
Example 2
Graph the following radicals by shifting the basic graph

f(x)=x+3f(x) = \sqrt{x + 3}
g(x)=−2+x3g(x) = -2 + \sqrt[3]{x}

ANSWER:

The graph of f(x)f(x) should be shifted to the left 3 units.

The graph of g(x)g(x) should be shifted down by 2 units.


Radicals and Trigonometry



When solving for the missing side of a right triangle we can use the Pythagorean Theorem.
Here the values of aa and bb represent the legs, and cc is the longest side or hypotenuse.

a2+b2=c2a^2+b^2=c^2

If we are looking for the value of one of the shorter legs, this equation can be rearranged to isolate the missing side.

a=c2−b2a = \sqrt{c^2-b^2}

Use this formula and the diagram below to help answer the questions.


1. What is the value of the missing leg in the triangle.

ANSWER:
Putting in the known values we have
a=172−152a=289−225a=64a=8\begin{aligned} a &= \sqrt{17^2 - 15^2} \\ a &= \sqrt{289- 225} \\ a &= \sqrt{64} \\ a &= 8 \end{aligned}


2. Suppose that the only known value was a leg fixed at 5cm. Draw a graph that show the relationship between aa and c2c^2.

ANSWER:
If the only known value is b=5b = 5, then from our formula we have
a=c2−25a = \sqrt{c^2 - 25}
From this we can draw a graph that shows the relationship between the two missing quantities.



3. What do points along this graph represent?

ANSWER:
The values along the horizontal axis represent values for c2c^2 and the values along the vertical axis represent values for aa. This means any given point along the graph gives us possible values for the missing sides of the triangle.

Match the Graph


Match each radical function with its graph.
A.


B.


C.


D.


f(x)=x−4f(x) = \sqrt{x -4}
f(x)=−4+x3f(x) = -4 + \sqrt[3]{x}
f(x)=x+4f(x) = \sqrt{x + 4}
f(x)=4+x3f(x) = 4 + \sqrt[3]{x}

Graphs of Radical Functions


Becky is walking in the park along a stream. To her right is a tree, as seen in the diagram.
The distance between Becky and the tree can be modeled with the following graph. Here the horizontal axis represents timett in seconds, and the vertical axis represents distance dd in feet.




1. When t=0t=0, how far away is Becky from the tree?

2. What equation best models the graph of this function?
1. When t=0t=0, how far away is Becky from the tree? (in feet)