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Answer :
Sure! Let's go through each question step-by-step:
Question 5: Complete the equation of the line that passes through (-2, 6) and (-1, 78) in slope-intercept form.
1. Calculate the slope (m):
- The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex].
- Here, the points are [tex]\((-2, 6)\)[/tex] and [tex]\((-1, 78)\)[/tex].
- Plug in the values: [tex]\( m = \frac{78 - 6}{-1 + 2} = \frac{72}{1} = 72 \)[/tex].
2. Find the y-intercept (b):
- With the slope-intercept form [tex]\( y = mx + b \)[/tex], use one of the points to find [tex]\( b \)[/tex].
- Using the point [tex]\((-2, 6)\)[/tex]: [tex]\( 6 = 72(-2) + b \)[/tex].
- Simplify to find [tex]\( b \)[/tex]: [tex]\( 6 = -144 + b \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( b = 150 \)[/tex].
3. Write the equation of the line:
- Substitute the values of the slope and y-intercept into the equation: [tex]\( y = 72x + 150 \)[/tex].
Question 6: The cost [tex]\( C \)[/tex] to rent a car for [tex]\( d \)[/tex] days is shown in the table. Which equation represents this function?
- Based on the answer provided, the equation that represents this function is [tex]\( C = 72d - 40 \)[/tex].
Question 7: Choose the function that represents the graph on the coordinate plane.
- The function that represents the graph on the coordinate plane is [tex]\( y = -4x \)[/tex].
Question 3: The graph of a function has a slope of -6, passes through the point (-3, 4), and is a straight line. Choose all the statements that must be true.
1. Calculate the y-intercept (b):
- Use the slope-intercept form [tex]\( y = mx + b \)[/tex] with the given point [tex]\((-3, 4)\)[/tex] and [tex]\( m = -6 \)[/tex].
- Plug in the values: [tex]\( 4 = -6(-3) + b \)[/tex].
- Simplify to find [tex]\( b \)[/tex]: [tex]\( 4 = 18 + b \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( b = -14 \)[/tex].
2. Evaluate the statements:
- A: The graph will pass through the point [tex]\((1, 20)\)[/tex].
- Substitute [tex]\( x = 1 \)[/tex] into the equation [tex]\( y = -6x - 14 \)[/tex].
- Compute [tex]\( y = -6(1) - 14 = -6 - 14 = -20 \)[/tex], not 20. This statement is false.
- B: The graph will pass through the point [tex]\((-1, 8)\)[/tex].
- Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( y = -6(-1) - 14 \)[/tex].
- Compute [tex]\( y = 6 - 14 = -8 \)[/tex], not 8. This statement is false.
- C: The graph represents a discrete function. (A straight line is continuous, not discrete. Thus, this is false.)
- D: The function has a y-intercept at [tex]\((0, -14)\)[/tex].
- We found that [tex]\( b = -14 \)[/tex]. This statement is true.
- E: The constant rate of change is -6.
- The slope is given as -6, showing the rate of change. This statement is true.
The results for Question 3 are that options D and E are true.
Question 5: Complete the equation of the line that passes through (-2, 6) and (-1, 78) in slope-intercept form.
1. Calculate the slope (m):
- The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex].
- Here, the points are [tex]\((-2, 6)\)[/tex] and [tex]\((-1, 78)\)[/tex].
- Plug in the values: [tex]\( m = \frac{78 - 6}{-1 + 2} = \frac{72}{1} = 72 \)[/tex].
2. Find the y-intercept (b):
- With the slope-intercept form [tex]\( y = mx + b \)[/tex], use one of the points to find [tex]\( b \)[/tex].
- Using the point [tex]\((-2, 6)\)[/tex]: [tex]\( 6 = 72(-2) + b \)[/tex].
- Simplify to find [tex]\( b \)[/tex]: [tex]\( 6 = -144 + b \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( b = 150 \)[/tex].
3. Write the equation of the line:
- Substitute the values of the slope and y-intercept into the equation: [tex]\( y = 72x + 150 \)[/tex].
Question 6: The cost [tex]\( C \)[/tex] to rent a car for [tex]\( d \)[/tex] days is shown in the table. Which equation represents this function?
- Based on the answer provided, the equation that represents this function is [tex]\( C = 72d - 40 \)[/tex].
Question 7: Choose the function that represents the graph on the coordinate plane.
- The function that represents the graph on the coordinate plane is [tex]\( y = -4x \)[/tex].
Question 3: The graph of a function has a slope of -6, passes through the point (-3, 4), and is a straight line. Choose all the statements that must be true.
1. Calculate the y-intercept (b):
- Use the slope-intercept form [tex]\( y = mx + b \)[/tex] with the given point [tex]\((-3, 4)\)[/tex] and [tex]\( m = -6 \)[/tex].
- Plug in the values: [tex]\( 4 = -6(-3) + b \)[/tex].
- Simplify to find [tex]\( b \)[/tex]: [tex]\( 4 = 18 + b \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( b = -14 \)[/tex].
2. Evaluate the statements:
- A: The graph will pass through the point [tex]\((1, 20)\)[/tex].
- Substitute [tex]\( x = 1 \)[/tex] into the equation [tex]\( y = -6x - 14 \)[/tex].
- Compute [tex]\( y = -6(1) - 14 = -6 - 14 = -20 \)[/tex], not 20. This statement is false.
- B: The graph will pass through the point [tex]\((-1, 8)\)[/tex].
- Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( y = -6(-1) - 14 \)[/tex].
- Compute [tex]\( y = 6 - 14 = -8 \)[/tex], not 8. This statement is false.
- C: The graph represents a discrete function. (A straight line is continuous, not discrete. Thus, this is false.)
- D: The function has a y-intercept at [tex]\((0, -14)\)[/tex].
- We found that [tex]\( b = -14 \)[/tex]. This statement is true.
- E: The constant rate of change is -6.
- The slope is given as -6, showing the rate of change. This statement is true.
The results for Question 3 are that options D and E are true.
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