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, 26 and 2, 34 and 4, 34 and 8, 30 and 10, 0 and 12, 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 14 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) = 0.2x + 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	86
2	84
3	82

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	$325
5	$400
10	$475

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 12 days? (2 points)

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

Part A: Given the function g(x) = |x − 7|, 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| + 2, 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: The water balloon's height is increasing between the ordered pairs (0, 26) and (2, 34). Therefore, the height of the water balloon is increasing during the interval $0 \leq x < 2$. ### Part B: The water balloon's height stays the same between the ordered pairs (2, 34) and (4, 34). Hence, the height of the water balloon is constant during the interval $2 \leq x < 4$. ### Part C: The height of the water balloon is decreasing the fastest between the ordered pairs (10, 0) and (8, 30). The change from 30 feet at 8 seconds to 0 feet at 10 seconds occurs over 2 seconds, resulting in a steep decline. Thus, the interval with the fastest decrease is from $8$ seconds to $10$ seconds, as the height drops from $30$ feet to $0$ feet, representing a significant drop in a short time. ### Part D: Based on the pattern from the given data, after 12 seconds, the height of the water balloon is predicted to remain at $0$ feet. The balloon will have already hit the ground at 10 seconds, and remains at $0$ feet beyond that time for any time intervals greater than or equal to 10 seconds. --- ## Question 2 ### Part A: To find the test average for the math class after completing test 2, we substitute $x = 2$ into the function: $$ f(2) = 0.2(2) + 79 = 0.4 + 79 = 79.4 $$ Thus, the average test score in math after test 2 is **79.4**. ### Part B: To determine the test average for the science class after completing test 2, we refer to the table where $g(2) = 84$. Therefore, the average test score in science after test 2 is **84**. ### Part C: To find the average score after completing test 4 for both math and science, we first calculate $f(4)$: $$ f(4) = 0.2(4) + 79 = 0.8 + 79 = 79.8 $$ For science, we need to identify $g(4)$. The pattern shows that the average test score decreases by 2 points per test: - $g(1) = 86$ - $g(2) = 84$ - $g(3) = 82$ - Therefore, $g(4) = 80$. Now, we compare the two scores after test 4: - **Math average = 79.8** - **Science average = 80** Hence, the science class had a higher average after completing test 4, with an average score of **80** compared to the math class's score of **79.8**. --- ## Question 3 ### Part A: To find the slope of the function, we can take two points from the table: - Point 1: (0, 325) - Point 2: (5, 400) Using the slope formula $(y_2 - y_1)/(x_2 - x_1)$: $$ \text{slope} = \frac{400 - 325}{5 - 0} = \frac{75}{5} = 15 $$ The slope of 15 means that the account balance increases by $15 for each day that passes. ### Part B: Using point-slope form, we use the point (0, 325): 1. Point-Slope form: $y - y_1 = m(x - x_1)$: $$ g(x) - 325 = 15(x - 0) \implies g(x) = 15x + 325 $$ 2. Slope-Intercept form: $g(x) = 15x + 325$. 3. Standard form: $15x - g(x) + 325 = 0$ (or, $15x - y + 325 = 0$). ### Part C: The equation using function notation is: $$ g(x) = 15x + 325 $$ ### Part D: To find the balance in the bank account after 12 days, we substitute $x = 12$ into the equation: $$ g(12) = 15(12) + 325 = 180 + 325 = 505 $$ The balance in the bank account after 12 days is **$505**. --- ## Question 4 ### Part A: The function $g(x) = |x - 7|$ is an absolute value function with a vertex at the point (7, 0). The domain of the function is all real numbers, expressed as $(-∞, ∞)$. The range is all non-negative real numbers, expressed as $[0, ∞)$. ### Part B: The transformation from $f(x) = |x|$ to $h(x) = |x| + 2$ indicates a vertical shift upwards by $2$ units. The vertex of $h(x)$ moves from (0, 0) for $f(x)$ to (0, 2) for $h(x)$. Consequently, the range of $h(x)$ becomes $[2, ∞)$ instead of $[0, ∞)$.