8-3 Qualitative Graphs

8-3 Qualitative Graphs

neighborhood. Describe the 1. The graph below displays the distance from Luis’ home as he walks his dog in his neighborhood. change in the distance from his home over time.

SOLUTION:

Starting at the origin, look at the relationship between time and the distance traveled. Note which sections of the graph increase, decrease, or show no change.

Sample answer: Luis starts out from his home. He walks away from his home, stops to let the dog run around, and walks further away from home. Then he walks towards home. 2. The graph below displays the speed of a city bus as it stops frequently to pick up passengers. Describe the change in the speed over time.

SOLUTION:

Look at the relationship between time and speed. Note which sections of the graph increase, decrease, or show no change.

Sample answer: The speed of the bus increases at a constant rate, then remains constant, and then slows down. As it is picking up passengers, the speed is zero. This pattern continues. e S o lu t io n s M a n u a l - P o w e r e d b y C o g n e r o

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3. A grand piano that is over 100 years old has increased in value rapidly from when it was first purchased. Sketch a qualitative graph to represent this situation.

SOLUTION:

Sample answer: Label the vertical axis “Value.” Label the horizontal axis “Time.” The origin represents the value when the grand piano was purchased. Sketch the shape of the graph. The value of the grand piano increases over time, so sketch a linear graph with a positive slope.

e S o lu t io n s M a n u a l - P o w e r e d b y C o g n e r o

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4. An athlete alternates between running and walking during a workout. Sketch a qualitative graph to represent this situation.

SOLUTION:

Sample answer: Label the vertical axis “Distance.” Label the horizontal axis “Time.” Sketch the shape of the graph. The distance the athlete travels increases over time, so sketch an increasing function. The athlete alternates between running and walking, traveling at a greater rate during running than walking.

5. Reason Abstractly Use the graph below which displays the rate at which Hector hiked along a path.

a. What situation could the horizontal line segment represent? b. What situation could the vertical line segment represent? c. Did Hector ’s rate increase or decrease during the first portion of his hike? Explain your reasoning. SOLUTION:

a. Sample answer: The graph shows the change in Hector's hiking rate over time. So, the horizontal line segment could represent the period of time that Hector hiked at a steady rate. b. Sample answer: The graph shows the change in Hector's hiking rate over time. So, the vertical line segment could represent Hector suddenly stopping during his hike. c. During the first portion of Hector's hike, the graph rises from left to right. So, Hector's rate increased at the beginn beginning ing of the hike. hike. e S o lu t io n s M a n u a l - P o w e r e d b y C o g n e r o

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6. Persevere with Problems Use the graph below which displays the speed of a car as time increases. Draw a qualitative graph that represents the distance the car travels as time increases.

SOLUTION:

Sample answer: The speed of the car increases at a constant rate, then remains constant, and then slows down at a constant rate. So, the distance the car travels increases over time until the car comes to a stop. Label the vertical axis “Distance.” Label the horizontal axis “Time.” Sketch the shape of the graph. As time increases, the distance increases at a varied rate and then levels off when the car stops.

7. Reason Inductively A tree grows steadily. When it reaches a specific height, it stops growing. Which graph displays this relationship? Explain your reasoning to a classmate.

SOLUTION:

Sample answer: Graph A increases from left to right at a constant rate then levels off. This represents a tree growing steadily before it stops growing. Graph B represents a tree growing steadily and then decreasing its height steadily. Graph C represents a tree that does not grow at first and then grows steadily. e S o lu t io n s M a n u a l - P o w e r e d b y C o g n e r o

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8. The graph represents the height of a rocket being launched.

Which statement is true? A. The graph is decreasing. B . The graph is increasing. C. The graph remains constant. D. The graph is linear SOLUTION:

The graphs shows that the height of the rocket increases at varied rates during launch. This is choice B. 9. The graph below displays the distance Justine rode on her bike. Describe the change in the distance over time.

SOLUTION:

Starting at the origin, look at the relationship between time and the distance traveled. Note which sections of the graph increase or show no change.

Sample answer: Justine rode her bike at a constant rate in the beginning. She then stopped riding for a period of time. Then she continued riding at a constant rate.


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10. The graph below displays the temperature of a cup of hot chocolate. Describe the change in the temperature over time.

SOLUTION:

Look at the relationship between time and temperature. Note which sections of the graph decrease or show no change.

Sample answer: The temperature cools down rapidly in the beginning. Then it cools down at a slower rate and levels off. 11. A graph of Mrs. Fraser ’s electric bill throughout the year is shown below. Describe the change in the bill over time.

SOLUTION:

Look at the relationship between time and the amount of the electric bill. Note which sections of the graph increase or decrease.

Sample answer: Mrs. Fraser ’s electric bill starts out high in January, increases until about March, and then decreases throughout the spring and summer. In the fall, the electric bill increases again.


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12. The graph below displays the distance covered on a long road trip. Describe the change in the distance over time.

SOLUTION:


Sample answer: The graph shows the car traveling at a constant speed then stopping, and then moving at a faster speed. The car stops a second time, then continues traveling.


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13. Model with Mathematics The outside temperature rises throughout the day at varied rates, then drops at night. Sketch a qualitative graph to represent the situation.

SOLUTION:

Sample answer: Label the vertical axis “Temperature. ” Label the horizontal axis “Time.” Sketch the shape of the graph. The temperature rises throughout the day at varied rates until it reaches the high temperature then begins to drop at night, so sketch a nonlinear increasing and decreasing function.


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14. A lion cub is resting in the grass. He sees another lion cub nearby and races after it, picking up speed as it runs. Sketch a qualitative graph to represent the situation.

SOLUTION:

Sample answer: Label the vertical axis “Speed.” Label the horizontal axis “Time.” The origin represents the time when the lion cub is resting. Sketch the shape of the graph. The speed of the lion cub increases at varied rates as it runs, so sketch a nonlinear increasing function.

15. The graph displays the number of pies sold by a bakery.

What does the red line segment represent? A. The number of pies sold increases at a constant rate. B . The number of pies sold decreases at a constant rate. C. The number of pies sold decreases, but not at a constant rate. D. The number of pies sold increases, but not at a constant rate. SOLUTION:

The red line segment represents a linear function with a positive slope. So, the number of pies sold increases at a constant rate. This is choice A.


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16. Short Response The graph represents the distance a basketball team runs during practice. Describe the change in the distance over time.

SOLUTION:


Sample answer: The basketball team starts by running or walking, and then stops for a short period of time, then continues at the same pace. Then they run or walk at a slower pace. 17. Marjorie purchased the items shown in the table.

a. Write an expression to show the total cost of the items. b. Suppose the shirt is $20. How much did she spend altogether? SOLUTION:

a. Write an expression that represents the cost of the shirt, skirt, and socks. Then simplify the expression.

b. Evaluate the expression for p = 20.

She spent $65 altogether.


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Simplify the expression. 18. 2( p p + 8) + 4 SOLUTION:

Use the Distributive Property to simplify the expression.

)(−3) + 9 19. (18 + t )( SOLUTION:


20. 30q(2) SOLUTION:

Use the Commutative and Associative Properties to simplify s implify the expression.

21. −5(n + 16) − 7 SOLUTION:



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8-3 Qualitative Graphs

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