Module 10 Graph Radical Functions

Learning Outcome

Graph radical functions using tables and transformations

You can also graph radical functions (such as square root functions) by choosing values for x and finding points that will be on the graph. Again, it is helpful to have some idea about what the graph will look like.

Think about the basic square root function, $f(x)=\sqrt{x}$ . Take a look at a table of values for x and y and then graph the function. Notice that all the values for x in the table are perfect squares. Since you are taking the square root of x, using perfect squares makes more sense than just finding the square roots of $0, 1, 2, 3, 4$ , etc.

x	f(x)
$0$	$0$
$1$	$1$
$4$	$2$
$9$	$3$
$16$	$4$

Recall that x can never be negative because when you square a real number, the result is always positive. For example, $\sqrt{49}$ , this means “find the number whose square is $49$ .” Since there is no real number that we can square and get a negative, the function $f(x)=\sqrt{x}$ will be defined for $x>0$ .

Take a look at the graph.

Curved line branching up and right from the point 0,0.

As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.

Multiplying $\sqrt{x}$ by a positive value changes the width of the half-parabola. Multiplying $\sqrt{x}$ by a negative number gives you the other half of a horizontal parabola.

In the following example, we will show how multiplying a radical function by a constant can change the shape of the graph.

Example

Match each of the following functions to the graph that it represents.

a) $f(x)=-\sqrt{x}$

b) $f(x)=2\sqrt{x}$

c) $f(x)=\dfrac{1}{2}\sqrt{x}$

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.

A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.

Show Solution

Function a) goes with graph 2)

Function b) goes with graph 1)

Function c) goes with graph 3)

Function a) $f(x)=-\sqrt{x}$ means that all the outputs will be negative – the function is the negative of the square roots of the input. This will give the other half of the parabola on its side. Therefore graph 2) goes with the function $f(x)=-\sqrt{x}$ .

Function b) $f(x)=2\sqrt{x}$ means take the square root of all the inputs, then multiply by two, so the outputs will be larger than the outputs for $\sqrt{x}$ . Graph 1) goes with the function $f(x)=2\sqrt{x}$ .

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.

Function c) $f(x)=\dfrac{1}{2}\sqrt{x}$ means take the squat=re root of the inputs then multiply by $\Large\frac{1}{2}$ . The outputs will be smaller than the outputs for $\sqrt{x}$ . Graph 3) goes with the function $f(x)=\dfrac{1}{2}\sqrt{x}$ .

A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.

Adding a value outside the radical moves the graph up or down. Think about it as adding the value to the basic y value of $\sqrt{x}$ , so a positive value added moves the graph up.

Example

Match each of the following functions to the graph that it represents.

a) $f(x)=\sqrt{x}+3$

b) $f(x)=\sqrt{x}-2$

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.

DM_U17_Final1stEd-11-12-12

Show Solution

Function a) goes with graph 2)

Function b) goes with graph 1)

Function a) $f(x)=\sqrt{x}+3$ means take the square root of all the inputs and add three, so the out puts will be greater than those for $\sqrt{x}$ , therefore graph 2) goes with this function.

DM_U17_Final1stEd-11-12-12

Function b) $f(x)=\sqrt{x}-2$ means take the square root of the input then subtract two. The outputs will be less than those for $\sqrt{x}$ , therefore graph 1) goes with this function.
DM_U17_Final1stEd-11-12-12

Adding a value inside the radical moves the graph left or right. Think about it as adding a value to x before you take the square root—so the y value gets moved to a different x value. For example, for $f(x)=\sqrt{x}$ , the square root is $3$ if $x=9$ . For $f(x)=\sqrt{x+1}$ , the square root is $3$ when $x+1$ is $9$ , so x is $8$ . Changing x to $x+1$ shifts the graph to the left by $1$ unit (from $9$ to $8$ ). Changing x to $x−2$ shifts the graph to the right by $2$ units.

Example

Match each of the following functions to the graph that it represents.

a) $f(x)=\sqrt{x+1}$

b) $f(x)=\sqrt{x-2}$

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0

Show Solution

Function a) matches graph 2)

Function b) matches graph 1)

Function a) $f(x)=\sqrt{x+1}$ adds one to the inputs before the square root is taken. The outputs will be greater, so it ends up looking like a shift to the left. Graph 2) matches this function.

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0

Function b) $f(x)=\sqrt{x-2}$ means subtract before the square root is taken. This makes the outputs less than they would be for the standard $\sqrt{x}$ , and looks like a shift to the right. Graph 1) matches this function.

A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0

Notice that as x gets greater, adding or subtracting a number inside the square root has less of an effect on the value of y.

In the next example, we will combine some of the changes that we have seen into one function.

Example

Graph $f(x)=-2+\sqrt{x-1}$ .

Show Solution

Before making a table of values, look at the function equation to get a general idea about what the graph should look like.

Inside the square root, you are subtracting $1$ , so the graph will move to the right $1$ from the basic $f(x)=\sqrt{x}$ graph.

You are also adding $−2$ outside the square root, so the graph will move down two from the basic $f(x)=\sqrt{x}$ graph.

Create a table of values. Choose values that will make your calculations easy. You want $x–1$ to be a perfect square ( $0, 1, 4, 9$ , and so on) so you can take the square root.

x	f(x)
$1$	$−2$
$2$	$−1$
$5$	$0$
$10$	$1$

Since values of x less than $1$ makes the value inside the square root negative, there will be no points on the coordinate graph to the left of $x=1$ . There is no need to choose x values less than $1$ for your table!

Use ordered pairs from each row of the table to plot points.

Connect the points as best you can using a smooth curve.

Review Topics for Success Summaries