Question

algebra i unit 2 retake notes - key features

1. distance the graph represents the distance y in miles roland is from home as a function of time x in hours. which could very likely be true? select all that apply.

the farthest distance roland is from home during these 12 hours is 4 miles.
roland returned home twice.
between hours 2 and 4, roland was 3 miles from home.
between hours 0 and 2, roland was getting farther from home at a faster rate than he was between hours 7 and 8.
roland spent 3 hours at home during these 12 hours.
rolands distance from home after 12 hours is greater than his distance from home after 6 hours.

1. the first roller coaster was said to have opened on june 16th, 1884 at coney island in brooklyn, new york. roller coasters have continued to improve and become more complex throughout the years. the graph below shows vertical height of a roller coaster versus the horizontal height of the coaster. use the graph to answer the questions below.

a. how many hills does the roller coaster have? b. place a star where the highest point is.
c. circle where the lowest point is on the coaster. how far from the start of the coaster does the rider reach this height?
d. highlight where the roller coaster is increasing in height.
e. use numbers and symbols to state where the coaster is increasing.

Explanation:

Step1: Analyze Roland's distance - time graph

For Roland's distance - time graph:

• The maximum value of the y - axis (distance) seems to be 4 miles, so the farthest distance Roland is from home during these 12 hours is 4 miles.
• The graph touches the x - axis (distance = 0) twice, so Roland returned home twice.
• Between hours 2 and 4, the y - value (distance) is constant at 3 miles.
• The slope between hours 0 and 2 is steeper than the slope between hours 7 and 8. The slope represents the rate of change of distance with respect to time, so Roland was getting farther from home at a faster rate between hours 0 and 2 than between hours 7 and 8.
• The graph is at y = 0 for a total of 3 non - overlapping time intervals, so Roland spent 3 hours at home during these 12 hours.
• After 12 hours, Roland is at home (y = 0), and after 6 hours, Roland is at a non - zero distance from home, so Roland's distance from home after 12 hours is not greater than his distance from home after 6 hours.

Step2: Analyze roller - coaster graph

a. A hill is a section that goes up and then down. Counting the number of such sections, the roller - coaster has 7 hills.
b. Visually identify the highest point on the graph and place a star there (this cannot be done in text format, but in practice, it would be the peak with the highest vertical height).
c. Visually identify the lowest point on the graph and circle it (again, not possible in text format). To find how far from the start, measure the horizontal distance (length of coaster) at that point. Let's assume the start is at 0 on the x - axis. If the lowest point is at x = 2200 ft (for example, depending on the actual graph), the rider reaches this height 2200 ft from the start.
d. Highlight the sections where the graph is going up (not possible in text format).
e. Let the x - axis represent the length of the coaster \(x\). If the coaster is increasing in height in the intervals \([0,100]\), \([1500,2000]\), \([2500,3000]\), \([3500,4000]\), \([4500,5000]\) (for example, depending on the actual graph), we can write the intervals as \(0\leq x\leq100\), \(1500\leq x\leq2000\), \(2500\leq x\leq3000\), \(3500\leq x\leq4000\), \(4500\leq x\leq5000\)

Answer:

1. The statements that are true:

• The farthest distance Roland is from home during these 12 hours is 4 miles.
• Roland returned home twice.
• Between hours 2 and 4, Roland was 3 miles from home.
• Between hours 0 and 2, Roland was getting farther from home at a faster rate than he was between hours 7 and 8.
• Roland spent 3 hours at home during these 12 hours.

2.
a. 7
b. (Star placement not possible in text)
c. (Circling not possible in text, but assume distance from start is some value depending on graph)
d. (Highlighting not possible in text)
e. Intervals where the coaster is increasing (example intervals given above depending on actual graph)