Question
Compare g(x)equals=negative 2 x squared minus 8 x plus 4−2x2−8x+4 to f, shown in the graph. Which function has a greater maximum value?
-5
5
-2
10
x
y
x y graph
Question content area bottom
Part 1
Choose the correct answer below.
A.
g left parenthesis x right parenthesisg(x); The x-coordinate of the vertex of g left parenthesis x right parenthesisg(x) is greater than the x-coordinate of the vertex of f left parenthesis x right parenthesisf(x).
B.
f left parenthesis x right parenthesisf(x); The x-coordinate of the vertex of f left parenthesis x right parenthesisf(x) is greater than the x-coordinate of the vertex of g left parenthesis x right parenthesisg(x).
C.
f left parenthesis x right parenthesisf(x); The y-coordinate of the vertex of f left parenthesis x right parenthesisf(x) is greater than the y-coordinate of the vertex of g left parenthesis x right parenthesisg(x).
D.
g left parenthesis x right parenthesisg(x); The y-coordinate of the vertex of g left parenthesis x right parenthesisg(x) is greater than the y-coordinate of the vertex of f left parenthesis x right parenthesisf(x).
-5
5
-2
10
x
y
x y graph
Question content area bottom
Part 1
Choose the correct answer below.
A.
g left parenthesis x right parenthesisg(x); The x-coordinate of the vertex of g left parenthesis x right parenthesisg(x) is greater than the x-coordinate of the vertex of f left parenthesis x right parenthesisf(x).
B.
f left parenthesis x right parenthesisf(x); The x-coordinate of the vertex of f left parenthesis x right parenthesisf(x) is greater than the x-coordinate of the vertex of g left parenthesis x right parenthesisg(x).
C.
f left parenthesis x right parenthesisf(x); The y-coordinate of the vertex of f left parenthesis x right parenthesisf(x) is greater than the y-coordinate of the vertex of g left parenthesis x right parenthesisg(x).
D.
g left parenthesis x right parenthesisg(x); The y-coordinate of the vertex of g left parenthesis x right parenthesisg(x) is greater than the y-coordinate of the vertex of f left parenthesis x right parenthesisf(x).
Answers
GPT-4o mini
To find out which function, \( g(x) = -2x^2 - 8x + 4 \) or \( f(x) \), has a greater maximum value, we first need to determine the maximum value of \( g(x) \) by finding the vertex of the parabola represented by this quadratic function.
### Step 1: Find the Vertex of \( g(x) \)
The formula for the x-coordinate of the vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a}
\]
For \( g(x) = -2x^2 - 8x + 4 \):
- \( a = -2 \)
- \( b = -8 \)
Plugging these values into the vertex formula gives:
\[
x = -\frac{-8}{2 \cdot -2} = \frac{8}{-4} = -2
\]
### Step 2: Find the y-coordinate of the Vertex of \( g(x) \)
Now we substitute \( x = -2 \) back into \( g(x) \) to find the maximum value:
\[
g(-2) = -2(-2)^2 - 8(-2) + 4
\]
\[
= -2(4) + 16 + 4
\]
\[
= -8 + 16 + 4
\]
\[
= 12
\]
### Step 3: Compare With \( f(x) \)
Now we need to evaluate the maximum value of \( f(x) \). Since the maximum value of \( f(x) \) is not provided, we can only conclude based on the answers given.
From our calculations:
- The maximum value of \( g(x) \) is **12**.
To compare it with \( f(x) \):
- If the maximum value of \( f(x) \) is less than **12**, then \( g(x) \) would have the greater maximum.
- If the maximum value of \( f(x) \) is greater than **12**, then \( f(x) \) would have the greater maximum.
### Final Answer Identification
Given the choices:
- A. \( g(x) \); The x-coordinate of the vertex of \( g(x) \) is greater than the x-coordinate of the vertex of \( f(x) \).
- B. \( f(x) \); The x-coordinate of the vertex of \( f(x) \) is greater than the x-coordinate of \( g(x) \).
- C. \( f(x) \); The y-coordinate of the vertex of \( f(x) \) is greater than the y-coordinate of \( g(x) \).
- D. \( g(x) \); The y-coordinate of the vertex of \( g(x) \) is greater than the y-coordinate of \( f(x) \).
Given that the y-coordinate of the vertex of \( g(x) \) is **12** and we want to know which function has the greater maximum value, if it’s implied that \( f(x) < 12 \), we would select option D.
Thus, if the maximum value of \( f(x) \) is truly less than 12, the correct answer would be:
**D. \( g(x) \); The y-coordinate of the vertex of \( g(x) \) is greater than the y-coordinate of the vertex of \( f(x) \).**
If possible, please confirm the maximum value of \( f(x) \) for a final decision.
### Step 1: Find the Vertex of \( g(x) \)
The formula for the x-coordinate of the vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a}
\]
For \( g(x) = -2x^2 - 8x + 4 \):
- \( a = -2 \)
- \( b = -8 \)
Plugging these values into the vertex formula gives:
\[
x = -\frac{-8}{2 \cdot -2} = \frac{8}{-4} = -2
\]
### Step 2: Find the y-coordinate of the Vertex of \( g(x) \)
Now we substitute \( x = -2 \) back into \( g(x) \) to find the maximum value:
\[
g(-2) = -2(-2)^2 - 8(-2) + 4
\]
\[
= -2(4) + 16 + 4
\]
\[
= -8 + 16 + 4
\]
\[
= 12
\]
### Step 3: Compare With \( f(x) \)
Now we need to evaluate the maximum value of \( f(x) \). Since the maximum value of \( f(x) \) is not provided, we can only conclude based on the answers given.
From our calculations:
- The maximum value of \( g(x) \) is **12**.
To compare it with \( f(x) \):
- If the maximum value of \( f(x) \) is less than **12**, then \( g(x) \) would have the greater maximum.
- If the maximum value of \( f(x) \) is greater than **12**, then \( f(x) \) would have the greater maximum.
### Final Answer Identification
Given the choices:
- A. \( g(x) \); The x-coordinate of the vertex of \( g(x) \) is greater than the x-coordinate of the vertex of \( f(x) \).
- B. \( f(x) \); The x-coordinate of the vertex of \( f(x) \) is greater than the x-coordinate of \( g(x) \).
- C. \( f(x) \); The y-coordinate of the vertex of \( f(x) \) is greater than the y-coordinate of \( g(x) \).
- D. \( g(x) \); The y-coordinate of the vertex of \( g(x) \) is greater than the y-coordinate of \( f(x) \).
Given that the y-coordinate of the vertex of \( g(x) \) is **12** and we want to know which function has the greater maximum value, if it’s implied that \( f(x) < 12 \), we would select option D.
Thus, if the maximum value of \( f(x) \) is truly less than 12, the correct answer would be:
**D. \( g(x) \); The y-coordinate of the vertex of \( g(x) \) is greater than the y-coordinate of the vertex of \( f(x) \).**
If possible, please confirm the maximum value of \( f(x) \) for a final decision.