Question
Question 1(Multiple Choice Worth 5 points)
(08.05 MC)
The quadratic functions f(x) and g(x) are described as follows:
f(x) = −4x2 + 5
x g(x)
0 0
1 1
2 5
3 1
4 0
Which of the following statements best compares the maximum value of the two functions?
The maximum value is the same for both functions.
f(x) has a greater maximum value than g(x).
g(x) has a greater maximum value than f(x).
The maximum values cannot be determined.
Question 2(Multiple Choice Worth 5 points)
(08.07 MC)
The table represents a quadratic function.
x y
−6 23
−5 8
−4 −1
−3 −4
−2 −1
−1 8
0 23
What is the equation of the function?
y = (x + 3)2 − 4
y = (x − 3)2 + 4
y = 3(x + 3)2 − 4
y = 3(x − 3)2 + 4
Question 3(Multiple Choice Worth 5 points)
(08.03 MC)
Solve for x. Round your answers to two decimal places.
2x2 + 7x = 3
x = 0.60 and x = −2.60
x = −0.60 and x = 2.60
x = 0.39 and x = −3.89
x = −0.39 and x = 3.89
Question 4(Multiple Choice Worth 5 points)
(08.05 MC)
The table below shows the values of f(x) and g(x) for different values of x. One of the functions is a quadratic function, and the other is an exponential function. Which function is most likely increasing quadratically?
x f(x) g(x)
1 3 3
2 6 9
3 11 27
4 18 81
5 27 243
f(x), because it grows faster than g(x)
g(x), because it will not intersect f(x)
g(x), because it grows slower than f(x)
f(x), because it grows slower than g(x)
Question 5(Multiple Choice Worth 5 points)
(08.02 MC)
Which of the following tables shows the correct steps to transform x2 + 8x + 15 = 0 into the form (x − p)2 = q?
[p and q are integers]
Step 1 x2 + 8x + 15 − 1 = 0 − 1
Step 2 x2 + 8x + 14 = −1
Step 3 (x + 4)2 = −1
Step 1 x2 + 8x + 15 − 2 = 0 − 2
Step 2 x2 + 8x + 13 = −2
Step 3 (x + 4)2 = −2
Step 1 x2 + 8x + 15 + 1 = 0 + 1
Step 2 x2 + 8x + 16 = 1
Step 3 (x + 4)2 = 1
Step 1 x2 + 8x + 15 + 2 = 0 + 2
Step 2 x2 + 8x + 17 = 2
Step 3 (x + 4)2 = 2
Question 6(Multiple Choice Worth 5 points)
(08.05 LC)
For f(x) = x2 and g(x) = (x − 5)2, in which direction and by how many units should f(x) be shifted to obtain g(x)?
Up 5 units
Down 5 units
Left 5 units
Right 5 units
Question 7(Multiple Choice Worth 5 points)
(08.01 LC)
A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water:
graph of y equals minus 2 times the square of x plus 20 times x plus 400
What do the x-intercepts of the graph represent?
There were 20 larvae in the water at 0 degrees Celsius.
There were 10 larvae in the water at 0 degrees Celsius.
The water had no larvae at −10 degrees Celsius and 20 degrees Celsius.
The water had maximum larvae at −10 degrees Celsius and 20 degrees Celsius.
Question 8(Multiple Choice Worth 5 points)
(08.04 MC)
Tabitha wants to hang a painting in a gallery. The painting and frame must have an area of 58 square feet. The painting is 7 feet wide by 8 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?
image of a frame 7 feet wide by 8 feet long, with an x on the bottom and right sides
x2 + 15x − 2 = 0
x2 + 15x + 58 = 0
4x2 + 30x − 2 = 0
4x2 + 30x + 58 = 0
Question 9 (Essay Worth 10 points)
(08.01 HC)
Use the function f(x) to answer the questions:
f(x) = 2x2 − 3x − 5
Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
Add Audio Add Video
Question 10 (Essay Worth 10 points)
(08.01, 08.04 MC)
The graph below shows a company's profit f(x), in dollars, depending on the price of erasers x, in dollars, sold by the company:
Graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 8, 0. The vertex is at 4, 270.
Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit? (4 points)
Part B: What is an approximate average rate of change of the graph from x = 1 to x = 4, and what does this rate represent? (3 points)
Part C: Describe the constraints of the domain. (3 points)
(08.05 MC)
The quadratic functions f(x) and g(x) are described as follows:
f(x) = −4x2 + 5
x g(x)
0 0
1 1
2 5
3 1
4 0
Which of the following statements best compares the maximum value of the two functions?
The maximum value is the same for both functions.
f(x) has a greater maximum value than g(x).
g(x) has a greater maximum value than f(x).
The maximum values cannot be determined.
Question 2(Multiple Choice Worth 5 points)
(08.07 MC)
The table represents a quadratic function.
x y
−6 23
−5 8
−4 −1
−3 −4
−2 −1
−1 8
0 23
What is the equation of the function?
y = (x + 3)2 − 4
y = (x − 3)2 + 4
y = 3(x + 3)2 − 4
y = 3(x − 3)2 + 4
Question 3(Multiple Choice Worth 5 points)
(08.03 MC)
Solve for x. Round your answers to two decimal places.
2x2 + 7x = 3
x = 0.60 and x = −2.60
x = −0.60 and x = 2.60
x = 0.39 and x = −3.89
x = −0.39 and x = 3.89
Question 4(Multiple Choice Worth 5 points)
(08.05 MC)
The table below shows the values of f(x) and g(x) for different values of x. One of the functions is a quadratic function, and the other is an exponential function. Which function is most likely increasing quadratically?
x f(x) g(x)
1 3 3
2 6 9
3 11 27
4 18 81
5 27 243
f(x), because it grows faster than g(x)
g(x), because it will not intersect f(x)
g(x), because it grows slower than f(x)
f(x), because it grows slower than g(x)
Question 5(Multiple Choice Worth 5 points)
(08.02 MC)
Which of the following tables shows the correct steps to transform x2 + 8x + 15 = 0 into the form (x − p)2 = q?
[p and q are integers]
Step 1 x2 + 8x + 15 − 1 = 0 − 1
Step 2 x2 + 8x + 14 = −1
Step 3 (x + 4)2 = −1
Step 1 x2 + 8x + 15 − 2 = 0 − 2
Step 2 x2 + 8x + 13 = −2
Step 3 (x + 4)2 = −2
Step 1 x2 + 8x + 15 + 1 = 0 + 1
Step 2 x2 + 8x + 16 = 1
Step 3 (x + 4)2 = 1
Step 1 x2 + 8x + 15 + 2 = 0 + 2
Step 2 x2 + 8x + 17 = 2
Step 3 (x + 4)2 = 2
Question 6(Multiple Choice Worth 5 points)
(08.05 LC)
For f(x) = x2 and g(x) = (x − 5)2, in which direction and by how many units should f(x) be shifted to obtain g(x)?
Up 5 units
Down 5 units
Left 5 units
Right 5 units
Question 7(Multiple Choice Worth 5 points)
(08.01 LC)
A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water:
graph of y equals minus 2 times the square of x plus 20 times x plus 400
What do the x-intercepts of the graph represent?
There were 20 larvae in the water at 0 degrees Celsius.
There were 10 larvae in the water at 0 degrees Celsius.
The water had no larvae at −10 degrees Celsius and 20 degrees Celsius.
The water had maximum larvae at −10 degrees Celsius and 20 degrees Celsius.
Question 8(Multiple Choice Worth 5 points)
(08.04 MC)
Tabitha wants to hang a painting in a gallery. The painting and frame must have an area of 58 square feet. The painting is 7 feet wide by 8 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?
image of a frame 7 feet wide by 8 feet long, with an x on the bottom and right sides
x2 + 15x − 2 = 0
x2 + 15x + 58 = 0
4x2 + 30x − 2 = 0
4x2 + 30x + 58 = 0
Question 9 (Essay Worth 10 points)
(08.01 HC)
Use the function f(x) to answer the questions:
f(x) = 2x2 − 3x − 5
Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
Add Audio Add Video
Question 10 (Essay Worth 10 points)
(08.01, 08.04 MC)
The graph below shows a company's profit f(x), in dollars, depending on the price of erasers x, in dollars, sold by the company:
Graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 8, 0. The vertex is at 4, 270.
Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit? (4 points)
Part B: What is an approximate average rate of change of the graph from x = 1 to x = 4, and what does this rate represent? (3 points)
Part C: Describe the constraints of the domain. (3 points)
Answers
Answered by
GPT-4o mini
Sure, let's go through the questions one by one!
### Question 1
**Answer**: **f(x) has a greater maximum value than g(x).**
- **Explanation**: The function \( f(x) = -4x^2 + 5 \) is a downward-opening parabola with a maximum point at \( x = 0 \), which gives a maximum value of 5. The values from the table for \( g(x) \) shows a peak at 2 (which is less than 5).
### Question 2
**Answer**: **y = (x + 3)² − 4**
- **Explanation**: The quadratic opens upwards and has its vertex at (−3,−4), which matches with the given data points.
### Question 3
**Answer**: **x = 0.60 and x = −2.60**
- **Explanation**: Rearranging \( 2x^2 + 7x - 3 = 0 \) and applying the quadratic formula yields these roots.
### Question 4
**Answer**: **f(x), because it grows faster than g(x)**
- **Explanation**: The values of \( f(x) \) increase by a linear rate, whereas \( g(x) \) doubles each time, indicating it grows exponentially.
### Question 5
**Answer**: **Step 1 x² + 8x + 15 + 1 = 0 + 1**
- **Explanation**: Completing the square for \( x^2 + 8x + 15 = 0 \) gives the third option a correct form.
### Question 6
**Answer**: **Right 5 units**
- **Explanation**: The transformation of \( f(x) = x^2 \) to \( g(x) = (x - 5)^2 \) indicates a translation to the right by 5 units.
### Question 7
**Answer**: **The water had no larvae at −10 degrees Celsius and 20 degrees Celsius.**
- **Explanation**: The x-intercepts show the temperatures at which the larvae population becomes zero.
### Question 8
**Answer**: **4x² + 30x + 58 = 0**
- **Explanation**: To find the thickness, we use the area of the entire rectangle including the frame, which leads to the quadratic equation.
### Question 9 (Essay)
#### Part A:
To find the x-intercepts of \( f(x) = 2x^2 - 3x - 5 \), set \( f(x) = 0 \).
Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4*2*(-5)}}{2*2} = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm 7}{4} \).
So, \( x = 2.5 \) and \( x = -1 \).
#### Part B:
The vertex can be found using \( x = -\frac{b}{2a} = -\frac{-3}{2(2)} = \frac{3}{4} \).
Calculating \( f(\frac{3}{4}) \) gives: \( f(\frac{3}{4}) = 2(\frac{3}{4})^2 - 3(\frac{3}{4}) - 5 \), which results in minimum value. Thus it's a minimum vertex.
#### Part C:
To graph \( f(x) \), find the x-intercepts and vertex from Part A and B. Plot these, and identify the axis of symmetry as \( x= \frac{3}{4} \). Sketch the graph using these points, labeling the axis and any key features.
### Question 10 (Essay)
#### Part A:
The x-intercepts (0, 0) and (8, 0) represent break-even points where no profit is made. The maximum value at (4, 270) represents the highest profit. The function is increasing from \( x = 0 \) to \( x = 4 \) (growing profit), and decreasing from \( x = 4 \) to \( x = 8 \) (decreasing profit).
#### Part B:
The approximate average rate of change from \( x = 1 \) to \( x = 4 \) can be calculated using the average slope formula for that interval, representing how profit changes with the price adjusted between these points.
#### Part C:
The constraints of the domain represent practical limits on prices and quantities, indicating prices can't be negative and typically can't exceed a certain threshold based on market demands and operations.
I hope this helps you understand the responses better! If you have any further questions, feel free to ask!
### Question 1
**Answer**: **f(x) has a greater maximum value than g(x).**
- **Explanation**: The function \( f(x) = -4x^2 + 5 \) is a downward-opening parabola with a maximum point at \( x = 0 \), which gives a maximum value of 5. The values from the table for \( g(x) \) shows a peak at 2 (which is less than 5).
### Question 2
**Answer**: **y = (x + 3)² − 4**
- **Explanation**: The quadratic opens upwards and has its vertex at (−3,−4), which matches with the given data points.
### Question 3
**Answer**: **x = 0.60 and x = −2.60**
- **Explanation**: Rearranging \( 2x^2 + 7x - 3 = 0 \) and applying the quadratic formula yields these roots.
### Question 4
**Answer**: **f(x), because it grows faster than g(x)**
- **Explanation**: The values of \( f(x) \) increase by a linear rate, whereas \( g(x) \) doubles each time, indicating it grows exponentially.
### Question 5
**Answer**: **Step 1 x² + 8x + 15 + 1 = 0 + 1**
- **Explanation**: Completing the square for \( x^2 + 8x + 15 = 0 \) gives the third option a correct form.
### Question 6
**Answer**: **Right 5 units**
- **Explanation**: The transformation of \( f(x) = x^2 \) to \( g(x) = (x - 5)^2 \) indicates a translation to the right by 5 units.
### Question 7
**Answer**: **The water had no larvae at −10 degrees Celsius and 20 degrees Celsius.**
- **Explanation**: The x-intercepts show the temperatures at which the larvae population becomes zero.
### Question 8
**Answer**: **4x² + 30x + 58 = 0**
- **Explanation**: To find the thickness, we use the area of the entire rectangle including the frame, which leads to the quadratic equation.
### Question 9 (Essay)
#### Part A:
To find the x-intercepts of \( f(x) = 2x^2 - 3x - 5 \), set \( f(x) = 0 \).
Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4*2*(-5)}}{2*2} = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm 7}{4} \).
So, \( x = 2.5 \) and \( x = -1 \).
#### Part B:
The vertex can be found using \( x = -\frac{b}{2a} = -\frac{-3}{2(2)} = \frac{3}{4} \).
Calculating \( f(\frac{3}{4}) \) gives: \( f(\frac{3}{4}) = 2(\frac{3}{4})^2 - 3(\frac{3}{4}) - 5 \), which results in minimum value. Thus it's a minimum vertex.
#### Part C:
To graph \( f(x) \), find the x-intercepts and vertex from Part A and B. Plot these, and identify the axis of symmetry as \( x= \frac{3}{4} \). Sketch the graph using these points, labeling the axis and any key features.
### Question 10 (Essay)
#### Part A:
The x-intercepts (0, 0) and (8, 0) represent break-even points where no profit is made. The maximum value at (4, 270) represents the highest profit. The function is increasing from \( x = 0 \) to \( x = 4 \) (growing profit), and decreasing from \( x = 4 \) to \( x = 8 \) (decreasing profit).
#### Part B:
The approximate average rate of change from \( x = 1 \) to \( x = 4 \) can be calculated using the average slope formula for that interval, representing how profit changes with the price adjusted between these points.
#### Part C:
The constraints of the domain represent practical limits on prices and quantities, indicating prices can't be negative and typically can't exceed a certain threshold based on market demands and operations.
I hope this helps you understand the responses better! If you have any further questions, feel free to ask!
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