Question
The quadratic function f(x)=-4x^2+8x+12 can be rewritten as both f(x)=-4(x-1)^2+16 and f(x)=-4(x-3)(x+1) Which statement is true about the quadratic function?
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is x = − 1.
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is x = − 1.
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
Answers
GPT-4o mini
Let's analyze the statements one by one regarding the quadratic function \( f(x) = -4x^2 + 8x + 12 \).
1. **The graph of the function has a minimum at (1, 16)**:
This statement is incorrect. Since the coefficient of \( x^2 \) is negative (\( -4 \)), the parabola opens downward, which means it has a maximum point, not a minimum. The vertex was correctly rewritten in the form \( f(x) = -4(x-1)^2 + 16 \), indicating that the maximum occurs at the point \( (1, 16) \). Thus, it is incorrect to claim it has a minimum.
2. **The graph of the function has a y-intercept at (0, 12)**:
To find the y-intercept of a function, we evaluate \( f(0) \):
\[
f(0) = -4(0)^2 + 8(0) + 12 = 12.
\]
Therefore, the y-intercept is indeed at \( (0, 12) \). This statement is true.
3. **The line of symmetry of the function is \( x = -1 \)**:
The line of symmetry for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). For our function:
- \( a = -4 \)
- \( b = 8 \)
\[
x = -\frac{8}{2(-4)} = 1.
\]
Thus, the line of symmetry is \( x = 1 \), making this statement incorrect.
4. **The function has zeros at \( (-3, 0) \) and \( (1, 0) \)**:
To find the zeros (roots) of the function, we set \( f(x) = 0 \):
\[
-4x^2 + 8x + 12 = 0.
\]
Factoring gives:
\[
-4(x-3)(x+1) = 0.
\]
The roots of the equation are \( x-3 = 0 \Rightarrow x = 3 \) and \( x+1 = 0 \Rightarrow x = -1 \). Thus, the zeros of the function are at \( (3, 0) \) and \( (-1, 0) \), making this statement incorrect.
In conclusion, the only true statement about the quadratic function is:
**The graph of the function has a y-intercept at (0, 12).**
1. **The graph of the function has a minimum at (1, 16)**:
This statement is incorrect. Since the coefficient of \( x^2 \) is negative (\( -4 \)), the parabola opens downward, which means it has a maximum point, not a minimum. The vertex was correctly rewritten in the form \( f(x) = -4(x-1)^2 + 16 \), indicating that the maximum occurs at the point \( (1, 16) \). Thus, it is incorrect to claim it has a minimum.
2. **The graph of the function has a y-intercept at (0, 12)**:
To find the y-intercept of a function, we evaluate \( f(0) \):
\[
f(0) = -4(0)^2 + 8(0) + 12 = 12.
\]
Therefore, the y-intercept is indeed at \( (0, 12) \). This statement is true.
3. **The line of symmetry of the function is \( x = -1 \)**:
The line of symmetry for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). For our function:
- \( a = -4 \)
- \( b = 8 \)
\[
x = -\frac{8}{2(-4)} = 1.
\]
Thus, the line of symmetry is \( x = 1 \), making this statement incorrect.
4. **The function has zeros at \( (-3, 0) \) and \( (1, 0) \)**:
To find the zeros (roots) of the function, we set \( f(x) = 0 \):
\[
-4x^2 + 8x + 12 = 0.
\]
Factoring gives:
\[
-4(x-3)(x+1) = 0.
\]
The roots of the equation are \( x-3 = 0 \Rightarrow x = 3 \) and \( x+1 = 0 \Rightarrow x = -1 \). Thus, the zeros of the function are at \( (3, 0) \) and \( (-1, 0) \), making this statement incorrect.
In conclusion, the only true statement about the quadratic function is:
**The graph of the function has a y-intercept at (0, 12).**