The linear model represents the height, f(x), of a water balloon thrown off the roof of a building over time, x, measured in seconds:

Question

A linear model with ordered pairs at 0, 60 and 2, 75 and 4, 75 and 6, 40 and 8, 20 and 10, 0 and 12, 0 and 14, 0. The x axis is labeled Time in seconds, and the y axis is labeled Height in feet.

Part A: During what interval(s) of the domain is the water balloon's height increasing? (2 points)

Part B: During what interval(s) of the domain is the water balloon's height staying the same? (2 points)

Part C: During what interval(s) of the domain is the water balloon's height decreasing the fastest? Use complete sentences to support your answer. (3 points)

Part D: Use the constraints of the real-world situation to predict the height of the water balloon at 16 seconds. Use complete sentences to support your answer. (3 points)

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Question 2 (Essay Worth 10 points)
(02.02, 02.05 MC)

The linear function f(x) = 0.9x + 79 represents the average test score in your math class, where x is the number of the test taken. The linear function g(x) represents the average test score in your science class, where x is the number of the test taken.

x	g(x)
1	78
2	79
3	80

Part A: Determine the test average for your math class after completing test 2. (2 points)

Part B: Determine the test average for your science class after completing test 2. (2 points)

Part C: Which class had a higher average after completing test 4? Show work to support your answer. (6 points)

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Question 3 (Essay Worth 10 points)
(02.02, 02.04 MC)

The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account:

x	g(x)
0	$1,500
2	$1,350
4	$1,200

Part A: Find and interpret the slope of the function. (3 points)

Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points)

Part C: Write the equation of the line using function notation. (2 points)

Part D: What is the balance in the bank account after 5 days? (2 points)

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Question 4 (Essay Worth 10 points)
(02.05 HC)

Part A: Given the function g(x) = |x − 5|, describe the graph of the function, including the vertex, domain, and range. (5 points)

Part B: If the parent function f(x) = |x| is transformed to h(x) = |x| + 3, what transformation occurs from f(x) to h(x)? How are the vertex and range of h(x) affected?

GPT-4o mini · Answer

## Question 1 ### Part A: During what interval(s) of the domain is the water balloon's height increasing? The water balloon's height is increasing between 0 seconds and 2 seconds. In this interval, the height increases from 60 feet (at x = 0) to 75 feet (at x = 2). ### Part B: During what interval(s) of the domain is the water balloon's height staying the same? The water balloon's height remains the same from 2 seconds until 4 seconds. The height is constant at 75 feet during this interval. ### Part C: During what interval(s) of the domain is the water balloon's height decreasing the fastest? The water balloon's height decreases the fastest between 4 seconds and 6 seconds, where it drops from 75 feet down to 40 feet. This is a decrease of 35 feet over 2 seconds, which corresponds to a vertical drop of 17.5 feet per second. ### Part D: Use the constraints of the real-world situation to predict the height of the water balloon at 16 seconds. By the time we reach 12 seconds, the water balloon has already reached a height of 0 feet, meaning it has hit the ground. Therefore, the height of the water balloon at 16 seconds will also be 0 feet, as it cannot be below ground level. --- ## Question 2 ### Part A: Determine the test average for your math class after completing test 2. To find the average after test 2, substitute x = 2 into the mathematical function: $$ f(2) = 0.9(2) + 79 = 1.8 + 79 = 80.8 $$ So, the average score in your math class after test 2 is **80.8**. ### Part B: Determine the test average for your science class after completing test 2. From the provided table for the science class: The average after test 2 is **79**. ### Part C: Which class had a higher average after completing test 4? Show work to support your answer. First, calculate the average score for the math class after test 4: $$ f(4) = 0.9(4) + 79 = 3.6 + 79 = 82.6 $$ Next, from the provided values for the science class, the average score at test 4 is not provided directly. We must assume a trend or interpolation. If we assume consistent increments based on the test scores: - Test 1: 78 - Test 2: 79 - Test 3: 80, We notice an increment of 1 point for each test. Thus, test 4's average for science: $$ g(4) = 80 + 1 = 81. $$ Now compare: - Math Average after Test 4: 82.6 - Science Average after Test 4: 81 The math class had a higher average after completing test 4 at **82.6** compared to the science class at **81**. --- ## Question 3 ### Part A: Find and interpret the slope of the function. To find the slope $ m $ of the function g(x) between any two points, we can use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Taking points (0, 1500) and (2, 1350): $$ m = \frac{1350 - 1500}{2 - 0} = \frac{-150}{2} = -75 $$ The slope of -75 indicates that for each day that passes, the bank account balance decreases by $75. ### Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. - **Point-Slope Form:** Using the point (0, 1500): $$ g(x) - 1500 = -75(x - 0) \rightarrow g(x) = -75x + 1500 $$ - **Slope-Intercept Form:** $$ g(x) = -75x + 1500 $$ - **Standard Form:** Rearranging: $$ 75x + g(x) = 1500 $$ It can also be written as: $$ 75x + y = 1500 $$ ### Part C: Write the equation of the line using function notation. Using function notation: $$ g(x) = -75x + 1500 $$ ### Part D: What is the balance in the bank account after 5 days? To find the balance after 5 days, substitute x = 5 into g(x): $$ g(5) = -75(5) + 1500 = -375 + 1500 = 1125 $$ The balance in the bank account after 5 days is **$1125**. --- ## Question 4 ### Part A: Given the function g(x) = |x − 5|, describe the graph of the function, including the vertex, domain, and range. The graph of $ g(x) = |x - 5| $ is a V-shape opening upwards. - **Vertex:** The vertex is at the point (5, 0) since this is where the function value is minimized. - **Domain:** The domain is all real numbers, $ (-\infty, \infty) $. - **Range:** The range is $ [0, \infty) $ because the absolute value function cannot be below zero. ### Part B: If the parent function f(x) = |x| is transformed to h(x) = |x| + 3, what transformation occurs from f(x) to h(x)? How are the vertex and range of h(x) affected? The transformation from $ f(x) = |x| $ to $ h(x) = |x| + 3 $ is a vertical shift upward by 3 units. - **Vertex:** The vertex of f(x) = |x| is at (0, 0). Therefore, the vertex of h(x) = |x| + 3 moves up to (0, 3). - **Range:** The range of the parent function $ f(x) $ is $ [0, \infty) $. The transformation shifts it to $ [3, \infty) $ for h(x). This indicates that the minimum value is now 3 due to the upward shift.