Review for Mastery amazonaws com
Name ________________________________________ Date __________________ Class__________________
Review for Mastery
LESSON
9-3
Graphing Quadratic Functions
You can use the axis of symmetry, vertex, and y-intercept to graph a quadratic function.
Step 3: Find the y-intercept.
Graph y = x2 + 6x + 8.
y=8
Step 1: Find the axis of symmetry. b 6 Use x = − = −3 x=− 2(1) 2a
Graph (0, 8). Step 4: Choose two x-values on the same side of the axis of symmetry as the point containing the y-intercept.
y = (0)2 + 6(0) + 8
Graph the axis of symmetry, x = −3. Step 2: Find the vertex. 2
Substitute 0 for x in the original equation. Simplify.
Use −2 and −1.
y = (−3) + 6(−3) + 8
Substitute −3 for x.
y = (−2)2 + 6 (−2) + 8 = 0 Graph (−2, 0). y = (−1)2 + 6 (−1) + 8 = 3 Graph (−1, 3).
y = 9 − 18 + 8 y = −1
Simplify.
Step 5: Reflect those points and connect them with a smooth curve.
Graph the vertex, (−3, −1).
1) axis of symmetry 3) y-intercept
2) vertex
Graph x2 + 4x − 12 by completing the following.
1. Find and graph the axis of symmetry. _________________________________________
2. Find and graph the vertex. _________________________________________
3. Find and graph the y-intercept. _________________________________________
4. Find and graph two more points. _________________________________________
5. Reflect the points and draw the graph.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
9-22
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class__________________ LESSON
9-3
Review for Mastery Graphing Quadratic Functions continued
Many real life situations involve quadratic functions. It is important to interpret the graphs correctly. The height in feet of a soccer ball kicked in the air can be modeled by the function f(t) = −16t 2 + 32t. Find the ball’s maximum height and the time it takes the ball to reach this height. Then find how long the ball is in the air.
The graph shows the approximate height of the soccer ball after t seconds. The x-axis is time t in seconds. The y-axis is the height h in feet. The vertex is the maximum height. 16 feet is the height at 1 second.
The height at 0 seconds is 0 feet.
The height at 2 seconds is 0 feet.
2 seconds is the length of time the ball is in the air. The maximum height is 16 feet. It takes the ball 1 second to reach this height. The soccer ball is in the air for 2 seconds. The height in feet of a rocket launched straight up in the air can be modeled by the function f(t) = −16t2 + 96t. The graph is shown.
6. Find the time it takes the rocket to reach the maximum height. ____________________________________
7. Find the rocket’s maximum height. ____________________________________
8. Find how long the rocket was in the air. ____________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
9-23
Holt McDougal Algebra 1
3. 18 feet; 1.5 seconds; 3 seconds
3.
4. Practice C
1.
5. 18 feet; 1 second; 2.5 seconds
2.
Review for Mastery
1. x = −2
2. (−2, −16)
3. (0, −12) 4. Possible answers: (−1, −15); (1, −7)
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A24
Holt McDougal Algebra 1
5.
2.
6. 3 seconds
7. 144 feet
36 ft; 1.5 seconds
8. 6 seconds
3. H 5. C
Challenge 1. 21; 11; 5; 3; 5; 11; 21
Reading Strategies 1. x = 3; (3, 4); (0, 5); Possible answers: (2, 3) and (1, 0)
3.18; 3.08; 3.02; 3.00; 3.02; 3.08; 3.18 2. (2, 3)
4. C
3. x = 2
4. 2 5. 2.27; 2.12; 2.03; 2.00; 2.03; 2.12; 2.27 6. (1, 2)
7. x = 1
8. 1 9. a. An equation for the axis of symmetry of b the graph of y = ax 2 + bx + c is x = . 2a b. One approach is to use graphs, and a second approach is to use tables as in Exercises 1 and 5.
2. x = 0; (0, 6); (0, 6); Possible answers: (2, 4) and (4, 2)
Problem Solving 1.
4 seconds Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A25
Holt McDougal Algebra 1
Review for Mastery
LESSON
9-3
Graphing Quadratic Functions
You can use the axis of symmetry, vertex, and y-intercept to graph a quadratic function.
Step 3: Find the y-intercept.
Graph y = x2 + 6x + 8.
y=8
Step 1: Find the axis of symmetry. b 6 Use x = − = −3 x=− 2(1) 2a
Graph (0, 8). Step 4: Choose two x-values on the same side of the axis of symmetry as the point containing the y-intercept.
y = (0)2 + 6(0) + 8
Graph the axis of symmetry, x = −3. Step 2: Find the vertex. 2
Substitute 0 for x in the original equation. Simplify.
Use −2 and −1.
y = (−3) + 6(−3) + 8
Substitute −3 for x.
y = (−2)2 + 6 (−2) + 8 = 0 Graph (−2, 0). y = (−1)2 + 6 (−1) + 8 = 3 Graph (−1, 3).
y = 9 − 18 + 8 y = −1
Simplify.
Step 5: Reflect those points and connect them with a smooth curve.
Graph the vertex, (−3, −1).
1) axis of symmetry 3) y-intercept
2) vertex
Graph x2 + 4x − 12 by completing the following.
1. Find and graph the axis of symmetry. _________________________________________
2. Find and graph the vertex. _________________________________________
3. Find and graph the y-intercept. _________________________________________
4. Find and graph two more points. _________________________________________
5. Reflect the points and draw the graph.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
9-22
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class__________________ LESSON
9-3
Review for Mastery Graphing Quadratic Functions continued
Many real life situations involve quadratic functions. It is important to interpret the graphs correctly. The height in feet of a soccer ball kicked in the air can be modeled by the function f(t) = −16t 2 + 32t. Find the ball’s maximum height and the time it takes the ball to reach this height. Then find how long the ball is in the air.
The graph shows the approximate height of the soccer ball after t seconds. The x-axis is time t in seconds. The y-axis is the height h in feet. The vertex is the maximum height. 16 feet is the height at 1 second.
The height at 0 seconds is 0 feet.
The height at 2 seconds is 0 feet.
2 seconds is the length of time the ball is in the air. The maximum height is 16 feet. It takes the ball 1 second to reach this height. The soccer ball is in the air for 2 seconds. The height in feet of a rocket launched straight up in the air can be modeled by the function f(t) = −16t2 + 96t. The graph is shown.
6. Find the time it takes the rocket to reach the maximum height. ____________________________________
7. Find the rocket’s maximum height. ____________________________________
8. Find how long the rocket was in the air. ____________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
9-23
Holt McDougal Algebra 1
3. 18 feet; 1.5 seconds; 3 seconds
3.
4. Practice C
1.
5. 18 feet; 1 second; 2.5 seconds
2.
Review for Mastery
1. x = −2
2. (−2, −16)
3. (0, −12) 4. Possible answers: (−1, −15); (1, −7)
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A24
Holt McDougal Algebra 1
5.
2.
6. 3 seconds
7. 144 feet
36 ft; 1.5 seconds
8. 6 seconds
3. H 5. C
Challenge 1. 21; 11; 5; 3; 5; 11; 21
Reading Strategies 1. x = 3; (3, 4); (0, 5); Possible answers: (2, 3) and (1, 0)
3.18; 3.08; 3.02; 3.00; 3.02; 3.08; 3.18 2. (2, 3)
4. C
3. x = 2
4. 2 5. 2.27; 2.12; 2.03; 2.00; 2.03; 2.12; 2.27 6. (1, 2)
7. x = 1
8. 1 9. a. An equation for the axis of symmetry of b the graph of y = ax 2 + bx + c is x = . 2a b. One approach is to use graphs, and a second approach is to use tables as in Exercises 1 and 5.
2. x = 0; (0, 6); (0, 6); Possible answers: (2, 4) and (4, 2)
Problem Solving 1.
4 seconds Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A25
Holt McDougal Algebra 1
