Question
Which of the following is true about the quadratic equation y = -2x2 + 4x + 6
A.
The quadratic equations has zeros at -1 and 3
B.
The quadratic equation has an axis of symmetry at x = -1
C.
The quadratic equation has a maximum at 8
D.
The quadratic equation has a minimum at 1
E.
The quadratic equation has a y intercept at (0,6)
A.
The quadratic equations has zeros at -1 and 3
B.
The quadratic equation has an axis of symmetry at x = -1
C.
The quadratic equation has a maximum at 8
D.
The quadratic equation has a minimum at 1
E.
The quadratic equation has a y intercept at (0,6)
Answers
GPT-4o mini
To analyze the quadratic equation \( y = -2x^2 + 4x + 6 \), we can evaluate each statement one by one.
1. **Finding the zeros**: To find the zeros, we set \( y = 0 \):
\[
-2x^2 + 4x + 6 = 0
\]
Dividing the entire equation by -2 to make calculations easier:
\[
x^2 - 2x - 3 = 0
\]
Factoring:
\[
(x - 3)(x + 1) = 0
\]
Therefore, the zeros are \( x = 3 \) and \( x = -1 \), which corrects statement A to be true.
2. **Axis of symmetry**: The axis of symmetry for a quadratic \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 4 \):
\[
x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1
\]
Therefore, statement B is false.
3. **Finding the vertex (maximum/minimum)**: The vertex is at the axis of symmetry \( x = 1 \). We substitute \( x = 1 \) into the equation to find the y value:
\[
y = -2(1^2) + 4(1) + 6 = -2 + 4 + 6 = 8
\]
Since the coefficient of \( x^2 \) is negative, this indicates a maximum. Thus, the maximum value is indeed 8, making statement C true.
4. **Minimum value**: Since the parabola opens downwards, there is no minimum value. Thus, statement D is false.
5. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
y = -2(0^2) + 4(0) + 6 = 6
\]
This means the y-intercept is \( (0, 6) \), making statement E also true.
In conclusion, the true statements are:
- A: True (zeros at -1 and 3)
- C: True (maximum at 8)
- E: True (y-intercept at (0, 6))
So, the answer is that statements A, C, and E are true.
1. **Finding the zeros**: To find the zeros, we set \( y = 0 \):
\[
-2x^2 + 4x + 6 = 0
\]
Dividing the entire equation by -2 to make calculations easier:
\[
x^2 - 2x - 3 = 0
\]
Factoring:
\[
(x - 3)(x + 1) = 0
\]
Therefore, the zeros are \( x = 3 \) and \( x = -1 \), which corrects statement A to be true.
2. **Axis of symmetry**: The axis of symmetry for a quadratic \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 4 \):
\[
x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1
\]
Therefore, statement B is false.
3. **Finding the vertex (maximum/minimum)**: The vertex is at the axis of symmetry \( x = 1 \). We substitute \( x = 1 \) into the equation to find the y value:
\[
y = -2(1^2) + 4(1) + 6 = -2 + 4 + 6 = 8
\]
Since the coefficient of \( x^2 \) is negative, this indicates a maximum. Thus, the maximum value is indeed 8, making statement C true.
4. **Minimum value**: Since the parabola opens downwards, there is no minimum value. Thus, statement D is false.
5. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
y = -2(0^2) + 4(0) + 6 = 6
\]
This means the y-intercept is \( (0, 6) \), making statement E also true.
In conclusion, the true statements are:
- A: True (zeros at -1 and 3)
- C: True (maximum at 8)
- E: True (y-intercept at (0, 6))
So, the answer is that statements A, C, and E are true.