Download Graphing Square Root Functions

ALGEBRA 1
“Graphing Square Root
Functions”
What is a “square
root function”?
Square Root Function: function, f(x), in which the independent variable, x, is inside
a radicand
Example: y = x (the simplest square root function) y =
x+3
y = 2x - 8
How do you find
all of the solutions
to a square root
function?
Since a radicand can’t be negative (you can’t square a negative value), the domain (x
values) is limited to those values of x that make the radicand greater than or equal to
zero. Therefore, you can create a function table to graph the solutions with x being
values that make the expression inside the radicand greater than or equal to zero. To
find the minimum value of the range, plug in the minimum value of x that will make the
expression in the radical equal zero.
Examples:
Subtract 3 or add 8 to both sides
Make a function table.
ALGEBRA 1
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.
Key Rules:
*The domain and range are the set of nonnegative real numbers.
*The x-intercept and the y-intercept are at (0, 0).
*The function is neither even nor odd.
*It is increasing on the interval (0, ).
*It has a minimum value of 0 at x = 0.
ALGEBRA 1
Graphing Square Root Functions
LESSON 10-5
Additional Examples
I
Find the domain of each function.
a. y =
x+5
x+5>
–0
Make the radicand >
– 0.
x>
– –5
The domain is the set of all numbers greater than or equal to –5.
b. y = 6
4x – 12
4x – 12 >
–0
Make the radicand >
– 0.
4x >
– 12
x>
– 3
The domain is the set of all numbers greater than or equal to 3.
ALGEBRA 1
Find the domain of the square-root function.
x– 4 ≥
0
x +3 ≥
–3
+4 +4
x
≥
4
The domain is the set
of all real numbers
greater than or equal
to 4.
x
0
–3
≥ –3
The domain is the
set of all real
numbers greater
than or equal to –3.
ALGEBRA 1
Find the domain of the square-root function.
2x – 1
+1
≥
0
+1
2x
≥
1
The domain is the set
of all real numbers
greater than or equal
to .
3x – 5
+5
3x
≥ 0
+5
≥ 5
The domain is the set
of all real numbers
greater than or equal
to .
ALGEBRA 1
“Graphing Square Root
Functions”
I
How do you graph
a square root
function?
To graph a square root function:
1. Make a function table with at least four x values. Make one of the x values the
minimum value of x that makes the expression inside the radicand equal zero.
2. Plot the x and y value on the coordinate grid.
3. Connect the points to form a curve (should look like half of a parabola)
Examples:
Step 1. Function Table
Step 2. Plot the points.
Step 3. Connect the points to form a half parabola shape
Notice there are no x-values to the left of 0 because the domain is x ≥ 0.
ALGEBRA 1
y a xh k
f ( x)  x
Add a positive number to x.
f ( x)  x  h Shift left h.
Add a negative number to x.
f ( x)  x  h Shift right h.
Add a positive
Add a negative number to number
the
toAdd
the radical.
a positive number to the
radical.
radical.
f ( x)  x  k
f ( x)  x  k
Up k.
Down k.
ALGEBRA 1
If a square-root function is given in one of these forms,
you can graph the parent function
and
translate it vertically or horizontally.
ALGEBRA 1
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.
ALGEBRA 1
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.
ALGEBRA 1
Graph each square-root function.
A.
B.
ALGEBRA 1
Graphing Square Root Functions
LESSON 10-5
Additional Examples
Graph y =
y=
x + 4 by translating the graph of
x.
For the graph y =
the graph of y =
x + 4,
x is shifted 4 units up.
ALGEBRA 1
Graphing Square Root Functions
LESSON 10-5
Additional Examples
Graph ƒ(x) =
y=
x + 3 by translating the graph of
x.
For the graph ƒ(x) =
the graph of y =
x + 3,
x is shifted to the left 3 units.
ALGEBRA 1
I
Graph
Step 1 Choose x-values greater
than or equal to 0 and generate
ordered pairs.
.
Step 2 Plot the points. Then
connect them with a smooth
curve.
x
0
4
1
7
4
10
6
11.35
ALGEBRA 1
Graph
.
Step 1 Choose x-values
greater than or equal to
0 and generate ordered
pairs.
x
0
1
4
6
Step 2 Plot the points.
Then connect them with
a smooth curve.
3
5
7
7.89
ALGEBRA 1
Graphing Square Root Functions
LESSON 10-5
1. Find the domain of the function ƒ(x) =
A. Graph y = 3
B. Graph y =
2x – 4.
x>2
x.
x – 3.
4. Describe how to translate the graph of y = x
to obtain the graph of the function y = x – 15.
Shift the graph to the right 15 units.
ALGEBRA 1
Graphing Square-root Functions
Graphing y  a x  k
Domain: all nonnegative numbers
Range: all numbers greater than or equal to k
1. y  2 x  1
Domain: x  0
Range: y  1
x
y
0 y  2 0  1 1
1 y  2 1  1 3
4 y  2 4  1 5
9 y  2 9  1 7
ALGEBRA 1
Graphing Square-root Functions
Graphing y  a x  k
2. y  2 x  1
Domain: x  0
Range: y  1
x
y
0 y  2 0  1 1
1 y  2 1  1 1
4 y  2 4  1 3
9 y  2 9  1 5
ALGEBRA 1
Graphing Square-root Functions
Graphing y  a x  k
3. y  2 x  2
Domain: x  0
Range: y  2
x
y
0 y  2 0  2  2
1 y  2 120
4 y  2 4 22
9 y  2 9 24
ALGEBRA 1
Graphing Square-root Functions
Graphing y  x  h
Domain: all numbers greater than or equal to h
Range: all nonnegative numbers
4. y  x  1
Domain: x  1
Range: y  0
x
y
1 y  11  0
2 y  2 1  1
5 y  51  2
10 y  10  1  3
ALGEBRA 1
Graphing Square-root Functions
Graphing y  x  h
5. y  x  1
Domain: x  1
Range: y  0
x
y
-1 y  1  1 0
0 y  0 1  1
3 y  31  2
8 y  8 1  3
ALGEBRA 1
Graphing Square-root Functions
Graphing y  x  h
6. y  x  4
Domain: x  4
Range: y  0
x
y
4 y  44 0
5 y  5 4  1
8 y  84  2
13 y  13  4 3
ALGEBRA 1
Graphing Square-root Functions
Compare the Graphs
y
x
0
1
4
9
x k
yya xxhhkk
y
0
1
2
3
y  3 x32
ALGEBRA 1
ALGEBRA 1