11-5Square-Root
11-5
Square-RootFunctions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
11
11-5 Square-Root Functions
Warm Up
Find each square root.
2.
6
1.
3.
–20
4.
12
undefined
Solve each inequality.
5. x + 5 ≥ 0 x ≥ –5 6. 0 ≤ 4x – 8 x ≥ 2
7. 0 ≤ 3x x ≥ 0
Holt Algebra 1
8. 10 – 3x ≥ 0
11-5 Square-Root Functions
Warm Up Continued
Compare. Write <, >, or =.
9.
7 <
10.
3 >
Holt Algebra 1
11-5 Square-Root Functions
Objectives
Identify square-root functions and
their domains and ranges.
Graph square-root functions.
Holt Algebra 1
11-5 Square-Root Functions
Vocabulary
square-root function
Holt Algebra 1
11-5 Square-Root Functions
The function
gives the speed in feet per
second of an object in free fall after falling x feet.
This function is different from others you have
seen so far. It contains a variable under the
square-root sign,
.
Holt Algebra 1
11-5 Square-Root Functions
Holt Algebra 1
11-5 Square-Root Functions
Example 1A: Evaluating Square-Root Functions
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it
has fallen 16 feet.
Write the speed function.
Substitute 16 for x.
= 8(4)
Simplify.
= 32
After an object has fallen 16 feet, its speed is 32 ft/s.
Holt Algebra 1
11-5 Square-Root Functions
Example 1B: Evaluating Square-Root Functions
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it has
fallen 20 feet. Round your answer to the nearest
tenth.
Write the speed function.
Substitute 20 for x. Use a
calculator to find the square
root.
= 8(4.47)
Simplify.
≈ 35.8
After an object has fallen 20 feet, its speed is about
35.8 ft/s.
Holt Algebra 1
11-5 Square-Root Functions
Check It Out! Example 1a
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it
has fallen 25 feet.
Write the speed function.
Substitute 25 for x.
= 8(5)
Simplify.
= 40
After an object has fallen 25 feet, its speed is 40 ft/s.
Holt Algebra 1
11-5 Square-Root Functions
Check It Out! Example 1b
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it has
fallen 15 feet. Round your answer to the nearest
hundredth.
= 8(3.87)
Write the speed function.
Substitute 15 for x. Use a
calculator to find the square
root.
≈ 30.98
Simplify.
After an object has fallen 15 feet, its speed is about
30.98 ft/s.
Holt Algebra 1
11-5 Square-Root Functions
Recall that the square root of a negative number
is not a real number. The domain (x-values) of a
square-root function is restricted to numbers that
make the value under the radical sign greater
than or equal to 0.
Holt Algebra 1
11-5 Square-Root Functions
Example 2A: Finding the Domain of Square-root
Functions
Find the domain of the square-root function.
x– 4 ≥
0
+4 +4
x
≥
4
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Add 4 to both
sides.
The domain is the set of all real numbers greater
than or equal to 4.
Holt Algebra 1
11-5 Square-Root Functions
Example 2B: Finding the Domain of Square-root
Functions
Find the domain of the square-root function.
x +3 ≥
–3
x
0
–3
≥ –3
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Subtract 3
from both sides.
The domain is the set of all real numbers greater than
or equal to –3.
Holt Algebra 1
11-5 Square-Root Functions
Check It Out! Example 2a
Find the domain of the square-root function.
2x – 1 ≥ 0
+1
+1
2x
≥ 1
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Add 1 to both
sides.
Divide both sides by 2.
The domain is the set of all real numbers greater
than or equal to .
Holt Algebra 1
11-5 Square-Root Functions
Check It Out! Example 2b
Find the domain of the square-root function.
3x – 5
+5
3x
≥ 0
+5
≥ 5
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Add 5 to both
sides.
Divide both sides by 3.
The domain is the set of all real numbers greater
than or equal to .
Holt Algebra 1
11-5 Square-Root Functions
The parent function for
square-root functions,
is graphed at
right. Notice there are no
x-values to the left of 0
because the domain is
x ≥ 0.
Holt Algebra 1
11-5 Square-Root Functions
If a square-root function is given in one of these
forms, you can graph the parent function
and translate it vertically or horizontally.
Holt Algebra 1
11-5 Square-Root Functions
Example 3A: Graphing Square-Root Functions
Graph
.
Since this function is in the
form f(x) =
, you can
graph it as a horizontal
translation of the graph of
f(x) =
Graph f(x) =
and then
shift the graph 3 units to the
right.
Holt Algebra 1
11-5 Square-Root Functions
Example 3B: Graphing Square-Root Functions
Graph
.
This is not a horizontal or vertical translation of
Step 1 Find the domain of the function.
x≥0
The expression under the radical sign
must be greater than or equal to 0.
The domain is the set of all real numbers greater
than or equal to 0.
Holt Algebra 1
.
11-5 Square-Root Functions
Example 3B Continued
Graph
.
Step 2 Choose x-values
greater than or equal to
0 and generate ordered
pairs.
x
0
4
1
7
4
10
6
11.35
Holt Algebra 1
Step 3 Plot the points.
Then connect them with
a smooth curve.
11-5 Square-Root Functions
Check It Out! Example 3a
Graph each square root function.
Since this function is in the
form f(x) =
, you can
graph it as a vertical
translation of the graph of
f(x) =
Graph f(x) =
and then
shift the graph 2 units up.
Holt Algebra 1
11-5 Square-Root Functions
Check It Out! Example 3b
Graph each square root function.
This is not a horizontal or vertical translation of
Step 1 Find the domain of the function.
x≥0
The expression under the radical sign
must be greater than or equal to 0.
The domain is the set of all real numbers greater
than or equal to 0.
Holt Algebra 1
.
11-5 Square-Root Functions
Check It Out! Example 3b Continued
Graph
.
Step 2 Choose x-values
greater than or equal to
0 and generate ordered
pairs.
x
0
3
1
5
4
7
6
7.89
Holt Algebra 1
Step 3 Plot the points.
Then connect them with
a smooth curve.
11-5 Square-Root Functions
Lesson Quiz: Part I
1. Use the formula
to find the radius of a circle
whose area is 28 in2. Use 3.14 for . Round your
answer to the nearest tenth of an inch. 3.0 in.
Find the domain of each square-root function.
2.
x≥0
3.
4.
Holt Algebra 1
x≥5
11-5 Square-Root Functions
Lesson Quiz: Part II
Graph each square-root function.
5.
Holt Algebra 1
6.