Question
Compare the function shown on the graph to f(x) = −
+ 8. Which statement is TRUE?
Responses
A The maximum value for both functions is the same.
The maximum value for both functions is the same.
B f(x) = −
+ 8 has a maximum value greater than the graph's function.f(x) = − x 2 + 8 has a maximum value greater than the graph's function.
C f(x) = −
+ 8 has a maximum value less than the graph's function.f(x) = − x 2 + 8 has a maximum value less than the graph's function.
D Both functions have a minimum value of 0Both functions have a minimum value of 0
+ 8. Which statement is TRUE?
Responses
A The maximum value for both functions is the same.
The maximum value for both functions is the same.
B f(x) = −
+ 8 has a maximum value greater than the graph's function.f(x) = − x 2 + 8 has a maximum value greater than the graph's function.
C f(x) = −
+ 8 has a maximum value less than the graph's function.f(x) = − x 2 + 8 has a maximum value less than the graph's function.
D Both functions have a minimum value of 0Both functions have a minimum value of 0
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the statements, we must first understand the function \( f(x) = -x^2 + 8 \).
This function is a quadratic function that opens downward (since the coefficient of \( x^2 \) is negative), and it has its vertex at the maximum point.
1. **Determine the maximum value of \( f(x) = -x^2 + 8 \)**:
- The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
- Here, the function can be rewritten as \( f(x) = -(x - 0)^2 + 8 \), which shows that the vertex is at \( (0, 8) \).
- Thus, the maximum value of \( f(x) \) is \( 8 \).
Now, let's evaluate the truth of each option based on the maximum value of the graph being referred to (assuming it's also a downward-opening parabola and that its maximum value needs to be compared with that of \( -x^2 + 8 \)).
- **Option A**: "The maximum value for both functions is the same."
- This statement is true only if the graph's maximum value is also 8.
- **Option B**: "f(x) = -x^2 + 8 has a maximum value greater than the graph's function."
- This statement would be true if the graph's maximum value is less than 8.
- **Option C**: "f(x) = -x^2 + 8 has a maximum value less than the graph's function."
- This statement can only be true if the graph's maximum value is greater than 8, which can't be since the standard quadratic forms are essentially capped based on their coefficients.
- **Option D**: "Both functions have a minimum value of 0."
- Since both functions open downward (if they are represented similarly), they don’t have a minimum value of 0; instead, they have a maximum above that point.
Thus, when we consider the maximum values derived from \( f(x) = -x^2 + 8 \), we find **Option A** to be true if the graph's maximum matches 8. Therefore, to finalize:
If we assume the "graph's function" is \( -x^2 + 8 \), **then Option A is true**. If it's a different function with a different maximum, we would need to know that specific function to assess the other options. But based on the information given and the nature of the comparison, A is the most compelling response.
This function is a quadratic function that opens downward (since the coefficient of \( x^2 \) is negative), and it has its vertex at the maximum point.
1. **Determine the maximum value of \( f(x) = -x^2 + 8 \)**:
- The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
- Here, the function can be rewritten as \( f(x) = -(x - 0)^2 + 8 \), which shows that the vertex is at \( (0, 8) \).
- Thus, the maximum value of \( f(x) \) is \( 8 \).
Now, let's evaluate the truth of each option based on the maximum value of the graph being referred to (assuming it's also a downward-opening parabola and that its maximum value needs to be compared with that of \( -x^2 + 8 \)).
- **Option A**: "The maximum value for both functions is the same."
- This statement is true only if the graph's maximum value is also 8.
- **Option B**: "f(x) = -x^2 + 8 has a maximum value greater than the graph's function."
- This statement would be true if the graph's maximum value is less than 8.
- **Option C**: "f(x) = -x^2 + 8 has a maximum value less than the graph's function."
- This statement can only be true if the graph's maximum value is greater than 8, which can't be since the standard quadratic forms are essentially capped based on their coefficients.
- **Option D**: "Both functions have a minimum value of 0."
- Since both functions open downward (if they are represented similarly), they don’t have a minimum value of 0; instead, they have a maximum above that point.
Thus, when we consider the maximum values derived from \( f(x) = -x^2 + 8 \), we find **Option A** to be true if the graph's maximum matches 8. Therefore, to finalize:
If we assume the "graph's function" is \( -x^2 + 8 \), **then Option A is true**. If it's a different function with a different maximum, we would need to know that specific function to assess the other options. But based on the information given and the nature of the comparison, A is the most compelling response.
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