To answer your questions, we need to analyze each function and their properties.
Question 1
The function given is \( f(x) = -x^2 + 8 \), which is a downward-opening parabola with its vertex (and thus maximum point) at the point (0,8). This means the maximum value of the function is 8.
Since you are comparing it to an unspecified graph (which we cannot see), we can only analyze the given function.
Now let's consider the answer choices:
- A: The maximum value for both functions is the same. (This can be true if the graph also has a maximum value of 8.)
- B: \( f(x) = -x^2 + 8 \) has a maximum value greater than the graph's function. (This can only be true if the graph's maximum value is less than 8.)
- C: \( f(x) = -x^2 + 8 \) has a maximum value less than the graph's function. (This can only be true if the graph's maximum value is greater than 8.)
- D: Both functions have a minimum value of 0. (This is not true since \( f(x) = -x^2 + 8 \) does not have a minimum value of 0; it opens downward.)
Given the information, without details about the graph, the best response would typically be option A, depending on the graph. However, if the graph's maximum value is indeed less than 8, then option B would hold.
Question 2
The function given is \( f(x) = x^2 + 2x + 3 \). To find its minimum value, we can complete the square or use the vertex formula. The vertex \( x \) coordinate is given by \( x = -\frac{b}{2a} \) where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \). Here we can see that:
- \( a = 1 \) and \( b = 2 \).
- Thus, \( x = -\frac{2}{2(1)} = -1 \).
We can find \( f(-1) \): \[ f(-1) = (-1)^2 + 2(-1) + 3 = 1 - 2 + 3 = 2. \] So, the minimum value of \( f(x) = x^2 + 2x + 3 \) is 2.
Now, comparing \( f(x) = x^2 + 2x + 3 \) to a function that has a minimum value of 2:
- A: The minimum value for both functions is 2. (This is true)
- B: The minimum value for \( f(x) = x^2 + 2x + 3 \) is greater than 2. (This is false)
- C: The minimum value for \( f(x) = x^2 + 2x + 3 \) is less than 2. (This is false)
- D: \( f(x) = x^2 + 2x + 3 \) has a maximum value of 2. (This is false; it has a minimum value of 2 and no maximum since it opens upwards)
The correct answer for Question 2 is A: The minimum value for both functions is 2.
In summary:
- For Question 1, the answer may vary based on the graph.
- For Question 2, the answer is A.
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