Question

from unit 5, lesson 8
5
sketch a graph to represent each quantity described as a function of time. be sure to label the vertical axis.
swing: the height of your feet above ground while swinging on a swing at a playground
slide: the height of your shoes above ground as you walk to a slide, go up a ladder, and then go down a slide
mery - go - round: your distance from the center of a merry - go - round as you ride the merry - go - round
merry - go - round, again: your distance from your friend, who is standing next to the merry - go - round as you go around
algebra 1
unit 5 • practice problems • illustrative mathematics®
cc by nc 2024

Explanation:

Step1: Analyze "Swing" Motion

For the swing: When you start swinging, your height above the ground starts at a minimum (when at rest or lowest point), then increases to a maximum as you swing up, decreases back to the minimum as you swing down, and this repeats periodically. So the graph should be a sinusoidal (wave - like) curve, starting from a non - zero minimum, going up and down in cycles.

Step2: Analyze "Slide" Motion

For the slide: First, when you walk to the slide, your height above the ground is constant (let's say at ground level or a small constant if walking on a flat surface). Then, as you go up the ladder, your height increases linearly (assuming a constant climbing speed) until you reach the top of the slide. Then, as you go down the slide, your height decreases linearly (assuming a constant sliding speed) back to the ground level. So the graph will have three parts: a horizontal line (walking), a rising linear segment (climbing ladder), and a falling linear segment (sliding down).

Step3: Analyze "Merry - go - round (your distance from center)"

When you ride the merry - go - round, your distance from the center of the merry - go - round is constant (equal to the radius of the circular path of the merry - go - round) as long as you are on the merry - go - round. So the graph is a horizontal line (constant distance) over time.

Step4: Analyze "Merry - go - round (your friend's distance from center)"

Your friend is standing next to the merry - go - round. So their distance from the center of the merry - go - round is constant (equal to the distance from their standing position to the center of the merry - go - round) and does not change over time. So the graph is a horizontal line (constant distance) over time.

(Note: Since the problem is about sketching graphs for different motion - related functions, the key is to analyze the behavior of each quantity (height or distance) with respect to time and then sketch the corresponding graph based on that behavior. If we were to actually draw the graphs:

• Swing Graph:
• Vertical axis: Height above ground. Horizontal axis: Time.
• Start at a value \(h_0\) (lowest height), then the height \(h(t)=h_0 + A\sin(\omega t+\phi)\) (where \(A\) is the amplitude of the swing, \(\omega\) is the angular frequency related to the swing's period, and \(\phi\) is the phase shift, usually \(\phi = 0\) if we start at the lowest point).
• Slide Graph:
• Vertical axis: Height above ground. Horizontal axis: Time.
• Let \(t_1\) be the time spent walking, \(t_2\) be the time spent climbing the ladder, and \(t_3\) be the time spent sliding down.
• For \(0\leq t\lt t_1\): \(h(t)=h_{ground}\) (horizontal line).
• For \(t_1\leq t\lt t_1 + t_2\): \(h(t)=h_{ground}+v_{climb}(t - t_1)\) (linear increase, where \(v_{climb}\) is the climbing speed).
• For \(t_1 + t_2\leq t\lt t_1 + t_2+t_3\): \(h(t)=h_{top}-v_{slide}(t-(t_1 + t_2))\) (linear decrease, where \(h_{top}\) is the height at the top of the slide and \(v_{slide}\) is the sliding speed).
• Merry - go - round (your distance) Graph:
• Vertical axis: Distance from center. Horizontal axis: Time.
• \(d(t)=r\) (constant, where \(r\) is the radius of the merry - go - round) for all \(t\) while on the ride.
• Merry - go - round (friend's distance) Graph:
• Vertical axis: Distance from center. Horizontal axis: Time.
• \(d(t)=d_0\) (constant, where \(d_0\) is the distance from the friend's position to the center) for all \(t\).)

Answer:

Step1: Analyze "Swing" Motion

For the swing: When you start swinging, your height above the ground starts at a minimum (when at rest or lowest point), then increases to a maximum as you swing up, decreases back to the minimum as you swing down, and this repeats periodically. So the graph should be a sinusoidal (wave - like) curve, starting from a non - zero minimum, going up and down in cycles.

Step2: Analyze "Slide" Motion

For the slide: First, when you walk to the slide, your height above the ground is constant (let's say at ground level or a small constant if walking on a flat surface). Then, as you go up the ladder, your height increases linearly (assuming a constant climbing speed) until you reach the top of the slide. Then, as you go down the slide, your height decreases linearly (assuming a constant sliding speed) back to the ground level. So the graph will have three parts: a horizontal line (walking), a rising linear segment (climbing ladder), and a falling linear segment (sliding down).

Step3: Analyze "Merry - go - round (your distance from center)"

When you ride the merry - go - round, your distance from the center of the merry - go - round is constant (equal to the radius of the circular path of the merry - go - round) as long as you are on the merry - go - round. So the graph is a horizontal line (constant distance) over time.

Step4: Analyze "Merry - go - round (your friend's distance from center)"

Your friend is standing next to the merry - go - round. So their distance from the center of the merry - go - round is constant (equal to the distance from their standing position to the center of the merry - go - round) and does not change over time. So the graph is a horizontal line (constant distance) over time.

(Note: Since the problem is about sketching graphs for different motion - related functions, the key is to analyze the behavior of each quantity (height or distance) with respect to time and then sketch the corresponding graph based on that behavior. If we were to actually draw the graphs:

• Swing Graph:
• Vertical axis: Height above ground. Horizontal axis: Time.
• Start at a value \(h_0\) (lowest height), then the height \(h(t)=h_0 + A\sin(\omega t+\phi)\) (where \(A\) is the amplitude of the swing, \(\omega\) is the angular frequency related to the swing's period, and \(\phi\) is the phase shift, usually \(\phi = 0\) if we start at the lowest point).
• Slide Graph:
• Vertical axis: Height above ground. Horizontal axis: Time.
• Let \(t_1\) be the time spent walking, \(t_2\) be the time spent climbing the ladder, and \(t_3\) be the time spent sliding down.
• For \(0\leq t\lt t_1\): \(h(t)=h_{ground}\) (horizontal line).
• For \(t_1\leq t\lt t_1 + t_2\): \(h(t)=h_{ground}+v_{climb}(t - t_1)\) (linear increase, where \(v_{climb}\) is the climbing speed).
• For \(t_1 + t_2\leq t\lt t_1 + t_2+t_3\): \(h(t)=h_{top}-v_{slide}(t-(t_1 + t_2))\) (linear decrease, where \(h_{top}\) is the height at the top of the slide and \(v_{slide}\) is the sliding speed).
• Merry - go - round (your distance) Graph:
• Vertical axis: Distance from center. Horizontal axis: Time.
• \(d(t)=r\) (constant, where \(r\) is the radius of the merry - go - round) for all \(t\) while on the ride.
• Merry - go - round (friend's distance) Graph:
• Vertical axis: Distance from center. Horizontal axis: Time.
• \(d(t)=d_0\) (constant, where \(d_0\) is the distance from the friend's position to the center) for all \(t\).)