RadF - Graphing Radical Functions Lesson

Graphing Radical Functions

Let's explore the graphs of radical functions. A radical function is any function that contains a variable inside a root (a radical). This includes square roots, cubed roots, or any nth root.

Here are some examples:

equation image indicator $LaTeX: y=\sqrt[]{\left(x+3\right)};\:y=\sqrt[3]{\left(x+17\right)};y=\sqrt[4]{z-1}$ $y=\sqrt[]{\left(x+3\right)};\:y=\sqrt[3]{\left(x+17\right)};y=\sqrt[4]{z-1}$

This does not include functions that only contain "numbers" inside the radicals. There must be independent variables inside the radicals.
Here are some examples of some non-radical functions: equation image indicator

$LaTeX: f\left(x\right)=x+\sqrt[]{2}\:and\:y=x^2+3x+\sqrt[]{5}$ $f\left(x\right)=x+\sqrt[]{2}\:and\:y=x^2+3x+\sqrt[]{5}$

Just as with polynomial and rational functions, graphs of these functions have a basic shape. We are going to explore two of the more common radical functions, the square root and the cube root. The graphs of equation image indicator $LaTeX: y=\sqrt[]{x}\:\:and\:\:y=\sqrt[3]{x}$ $y=\sqrt[]{x}\:\:and\:\:y=\sqrt[3]{x}$ are examples of radical functions.

Domain and Range

In graphing square root functions, you cannot use imaginary numbers. Therefore, the domain of a square root function will always be limited to values that make the expression in the root non-negative (zero can be used). Numbers that make the expression in the root negative would be excluded values.

The domain of a function f(x) is the set of all values of x for which f(x) is defined. The range of a function f(x) is the set of all values of f(x), where x is the domain of f.

For odd numbered radicals both the domain and range span all real numbers. For even numbered radical functions, the term inside the radical must be at or above zero, otherwise it is undefined. This means that only the x values that make the term inside an even numbered radical positive are defined and in the domain.

Graphing the Square Root Function

We have discussed the domain and range of a function in previous units and lessons. The domain of a function is the set of all the independent values (or input vales) of a function, and the range of a function is the set of all dependent values (or output values) of a function. Graphing the simplest radical function

equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$ , is almost as simple as finding the domain and range. The domain and range are easy to find for this function. Since x is the only thing inside the radical, x must be great than or equal zero (≥ 0). Plugging in this domain into the function, f(x) will also have all the values greater than or equal to zero (≥ 0). This means that the graph will lie only in the first quadrant, since neither x nor y can be negative. After substituting the first few known points an easy pattern starts to emerge. (0, 0), (1, 1), (4, 2) and (9, 3). Both the x's and y's are increasing, but the x's are increasing faster then the y's.

Graphs of radical functions also have intercepts, both x-intercepts (where the graph crosses the x-axis) and y-intercepts (where the graph crosses the y-axis). Graphs of radical functions do not always have intercepts, as not all radical functions cross the x-axis and or y-axis.

From this we can infer that the slope of the curve is increasing, and fitting the curve through the first known points results in a graph that looks like this:

We already know from our domain and range that the line will increase forever in both the x and y directions, so there are no asymptotes. This basic curve is the starting point for graphing all square root functions.

Translation of Radical Function Graphs

The basic form for a square root function is equation image indicator $LaTeX: f\left(x\right)=a\sqrt[]{x-h}+k$ $f\left(x\right)=a\sqrt[]{x-h}+k$ . A graph of a function in this form will start at the point (h, k) and will be the same line as $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$ only the point on the origin is translated from (0, 0) to (h, k).

Look at this example: equation image indicator $LaTeX: f\left(x\right)=a\sqrt[]{x+1}+1$ $f\left(x\right)=a\sqrt[]{x+1}+1$ .

Here h = -1 and k = 1. This means that the line will start from the point (-1, 1) and it will continue on the same path as equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$ .

Reflections of Radical Function Graphs

When a negative sign is added to the front of the radical, it will reflect the graph over the line y = b. This will leave the graph with the same origin and shape, but it will be "turned upside down." The graph of the line equation image indicator $LaTeX: f\left(x\right)=-\sqrt[]{x}$ $f\left(x\right)=-\sqrt[]{x}$ . It looks just like the graph of $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$ , but it is reflected over the line y = 0.

Graph of $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$

graph of y = squareroot x

Solution: Graph of $LaTeX: f\left(x\right)=-\sqrt[]{x}$ $f\left(x\right)=-\sqrt[]{x}$ text annotation indicator Solution: $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$

If a negative sign is put in front of the independent variable inside the radical, then the graph is reflected over the line x = h. This will switch the domain around and the domain will go from equation image indicator $LaTeX: \left(-\infty,\:h\right]$ $\left(-\infty,\:h\right]$ . The curve will look the same, except it will be "backwards."

Example

equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{-x}$ $f\left(x\right)=\sqrt[]{-x}$

text annotation indicator Graph of f(x) = squareroot(-x)

text annotation indicator Solution: $LaTeX: f\left(x\right)=\sqrt[]{-x}$ $f\left(x\right)=\sqrt[]{-x}$ Solution: $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$

Example

equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{x-3}+4$ $f\left(x\right)=\sqrt[]{x-3}+4$

Since LaTeX: x-3 $x-3$ is inside the radical, and the domain lies on all the points where x makes $x-3$ greater than or equal to zero.

$LaTeX: x-3\:\ge0 \\ x\ge 3$ $x-3\:\ge0 \\ x\ge 3$

This occurs at any point at or above three, so the domain of the function is [3, $LaTeX: \infty$ $\infty$ ) equation image indicator . The range of the function is then all the points of the y-axis that x hits for the given values of the domain. We start at the point LaTeX: x = 3 $x = 3$ and plug it into the equation.

equation image indicator $LaTeX: f\left(3\right)=\sqrt[]{\left(3\right)-3}+4 \\ f\left(3\right)=\sqrt[]{0}+4 \\ f\left(3\right)=4$ $f\left(3\right)=\sqrt[]{\left(3\right)-3}+4 \\ f\left(3\right)=\sqrt[]{0}+4 \\ f\left(3\right)=4$

It is then easy to see that by plugging in any number larger than 3 for x will result in an f(x) larger than 4, so the smallest number in the range is 4. It is not easy to see that the range will go to infinity, but it will. For any number above 4 that is set as the answer to the above function, there will also be an x to define it. Therefore, the range is [4, $LaTeX: \infty$ $\infty$ ) equation image indicator .

Square Root Functions

Example

Complete the table of values for the function, equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{x}$ $f\left(x\right)=\sqrt[]{x}$ . This is the square root function.

$chart with undefined solutions$

What did you notice about some of the values?

If you typed the function into a calculator and tried to evaluate it for some of the x-values, what message appeared? Why?

Not every x-value will produce a y-value. You cannot take the square root of negative numbers so some of the values will yield an error message on the calculator. Try graphing the function and identifying the domain and range on your own. How did you do? Check your answers compared to the following:

Solution:

What is the domain of this function? D[0, infinite)
What is the range of this function? R[0, infinite)
Image of graph with a positive slightly curved line

Example

Complete the table of values for the function, equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{x}+2$ $f\left(x\right)=\sqrt[]{x}+2$ .

$chart of values for the function square root of x plus 2$

What did you notice about some of the values?

If you typed the function into a calculator and tried to evaluate it for some of the x-values, what message appeared? Why?

Not every x-value will produce a y-value. You can't take the square root of negative numbers so some of the values will yield an error message on the calculator. This time though, not all the negative x-values produced an error message. Try graphing the function and identifying the domain and range on your own. How did you do? Check your answers compared to the following:

Solution:

What is the domain of this function? D[-2, infinite)
What is the range of this function? R[-2, infinite)
Image of graph with slightly curved line

Example

Complete the table of values for the function, equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{9-x^2}$ $f\left(x\right)=\sqrt[]{9-x^2}$ .

$table of values for the function square root of quantitiy nine minus x squared$

What did you notice about some of the values?

If you typed the function into a calculator and tried to evaluate it for some of the x-values, what message appeared? Why?

Solution:

What is the domain of this function: D[-3, 3)
What is the range of this function R[2, 3)
Image of graph with an arch

Now, let's determine the domain of this function, equation image indicator $LaTeX: f\left(x\right)=\sqrt[]{2x+5}$ $f\left(x\right)=\sqrt[]{2x+5}$ , without graphing it. Check your solution by graphing it on a graphing calculator. What did you come up with? Check your answers compared to the solution:

Solution: You should see that setting up the inequality $LaTeX: 2x+5\ge0$ $2x+5\ge0$ will help you determine the domain. Once you solve the inequality you will see that the domain must be $LaTeX: x\ge\left(-\frac{5}{2}\right)$ $x\ge\left(-\frac{5}{2}\right)$ . If you graph this function, you will see that the graph starts at (-2.5, 0) and extends in the positive direction.

Cube Root Functions

Now let's look at another common radical function, the cube root.

Complete the table of values for the function equation image indicator $LaTeX: f\left(x\right)=\sqrt[3]{x}$ $f\left(x\right)=\sqrt[3]{x}$ .

$table of values for the function cubed root of x$

Do you get any of the same error messages for this function that you did in the table of values for the square root function? Why do you think that is so?

There are no error messages this time. This is because you can take a cube root of a negative number, so the domain of this function is all real numbers. Try graphing the function and identifying the domain and range on your own. How did you do? Check your answers compared to the solution:

Solution:

What is the domain of this function: D[-infinite, infinite)
What is the range of this function R[-infinite, infinite)
Image of graph with an curved line

Pendulum

Tommy visited the Museum of History and Technology with his class. They saw Focault's Pendulum in Pendulum Hall and it was fascinating to Tommy. He knew from science class that the time it takes a pendulum to complete a full cycle or swing depends upon the length of the pendulum.

The formula is given by equation image indicator $LaTeX: T=2\Pi\sqrt[]{\frac{L}{32}}$ $T=2\Pi\sqrt[]{\frac{L}{32}}$ where T represents the time in seconds and L represents the length of the pendulum in feet. He timed the swing of the pendulum with his watch and found that it took about 8 seconds for the pendulum to complete a full cycle. Help him figure out the length of the pendulum in feet. What did you come up with?

Solution: $LaTeX: 8=2\Pi\sqrt[]{\frac{L}{32}}$ $8=2\Pi\sqrt[]{\frac{L}{32}}$ , , therefore the length is approximately 52 feet.

Tommy thought that a pendulum that took a full 20 seconds to complete a full cycle would be very dramatic for a museum. How long must that pendulum be? If ceilings in the museum are about 20 feet high, and 1 story is 20 feet high, would this pendulum be possible?

Solution: $LaTeX: 20=2\Pi\sqrt[]{\frac{L}{32}}$ $20=2\Pi\sqrt[]{\frac{L}{32}}$ , therefore the length is approximately 424 feet. The building would have to be over 16 stories tall to accommodate this pendulum!

Graphing Radical Functions Practice

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