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Accel. AG
Unit 9 – Radical Functions
Name _______________________________
Radical Functions and Transformations
1. Complete the table of values for the function f ( x)  x . This is the parent square root function.
0
1
2
3
4
x
3
4
2
1
f (x )
0
1
1.41
1.73
2
a. 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?
The square root of a negative number is imaginary
so the calculator gives an error message. We cannot
plot these points on a Cartesian plane.
b. Graph the function by plotting the points from the
table above.
c. What is the domain of this function?
[0, ∞)
d. What is the range of this function?
[0, ∞)
2. Complete the table of values for the function g ( x)  x  2 .
0
1
2
3
4
x
3
4
2
1
g (x )
0
1
1.41
1.73
2
2.24
2.45
a. 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?
The square root of a negative number is imaginary
so the calculator gives an error message. We cannot
plot these points on a Cartesian plane.
b. Graph the function by plotting the points from the
table above.
c. What is the domain of this function?
[−2, ∞)
d. What is the range of this function?
[0, ∞)
e. How does this function compare to the parent
function in #1?
This function has been translated to the left 2 units.
3. Complete the table of values for the function h( x)  9  x 2 .
0
1
2
3
4
x
3
4
2
1
h(x)
0
2.24
2.83
3
2.83
2.24
0
a. 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?
The square root of a negative number is imaginary
so the calculator gives an error message. We cannot
plot these points on a Cartesian plane.
b. Graph the function by plotting the points from the
table above.
c. What is the domain of this function?
[−3, 3]
d. What is the range of this function?
[0, 3]
e. How does this function compare to the function in #1?
This function does not have the same shape as the function in #1. This function looks like a
semi-circle.
4. Using the three examples above, make a conjecture about the domain of a radical function. A
conjecture is an opinion or conclusion formed on the basis of incomplete information.
Because the square root of a negative number is not real, we must determine the values of x that
make the radicand greater than or equal to 0.
5. Use your conjecture to determine the domain of the function k ( x)  2 x  5 , without using a
graphing calculator.
2𝑥 + 5 ≥ 0
2𝑥 ≥ −5
5
𝑥 ≥ −2
6. Check your solution above by graphing k(x) on a graphing calculator.
The left most point on the function is (–2.5, 0) and the x values increase from there. So the domain is
5
[−2.5, ∞) or 𝑥 ≥ − 2.
Transformations of Square Root Functions
f ( x)  a bx  h  k
Use the Geogebra applet to explore different transformations of the square root function. Record your
findings in the table below. Be specific about values and directions.
Parameter
a
Transformation
If |𝑎| > 1, the function is vertically stretched by a factor of a.
If 0 < |𝑎| < 1, the function is vertically compressed by a factor of a.
If a < 0, the function is reflected over the x-axis.
h
If b = 1 and h > 0, the function is translated right h units.
If b = 1 and h < 0, the function is translated left h units.
k
If k > 0, the function is translated up k units.
If k < 0, the function is translated down k units.
b
If |𝑏| > 1, the function is horizontally compressed by a factor of b.
If 0 < |𝑏| < 1, the function is horizontally stretched by a factor of b.
If b < 0, the function is reflected over the y-axis.
7. How can we use the transformations of a square root function to help us determine the domain and
range of the function?
Translations help us determine the starting or ending points of an even root function. Reflections
help us determine if an even root function increases or decreases without bound. Transformations
help us determine the endpoints in our domain for even root functions. Odd root functions have
domain and range of all real numbers.
8. Without graphing, determine the domain and range of each function.
a.
f ( x)   x  3
b. g ( x)  2  4 x  9
1
Domain: [−3, ∞)
Range: (−∞, 0]
Domain: (−∞, 2]
Range: [9, ∞)
9. Without graphing, determine the domain of each function.
a.
f ( x)  16  x 2
Domain: [−4, 4]
b. g ( x)  x 2  25
Domain: (−∞, ∞)
10. Complete the table of values for the function f ( x)  3 x . This is the parent cube root function.
0
1
2
6
8
x
8
6
2
1
f (x )
–2
–1.82
–1.26
–1
0
1
1.26
1.82
2
a. Do you get the same error messages for this function that you did in the table of values for the
square root functions? Why or why not?
No, the cube root of a negative number is negative
and the cube root of a positive number is positive.
b. Graph the function by plotting the points from the
table above.
c. What is the domain of this function?
(−∞, ∞)
d. What is the range of this function?
(−∞, ∞)
11. Complete the table of values for the function g ( x)  23 x  1 .
0
1
x
8
6
2
1
g (x )
–4.16
–3.83
–2.88
–2.52
–2
0
a. Graph the function by plotting the points from the
table above.
2
2
6
3.42
8
3.83
b. What is the domain of this function?
(−∞, ∞)
c. What is the range of this function?
(−∞, ∞)
d. How does this function compare to the parent
function in #10?
This function has been translated to the right 1 unit
and vertically stretched by a factor of 2.
12. Refer to the conjecture you made in #4. Does your conjecture about the domain of radical functions
hold true for all radical functions? Why or why not?
No. For any even root radical, the conjecture holds true because the even root of a negative number
is not real. For any odd root radical, the domain is all real numbers.