Question
Which statement is false about the quadratic function in standard form?
y=2x2+32x+124
(1 point)
Responses
The y-intercept is (0,124)
.
The y-intercept is left parenthesis 0 comma 124 right parenthesis .
The vertex of the quadratic is (−8,−4)
.
The vertex of the quadratic is left parenthesis negative 8 comma negative 4 right parenthesis .
The vertex of the quadratic is (8,−4)
.
The vertex of the quadratic is left parenthesis 8 comma negative 4 right parenthesis .
The quadratic function in vertex form is y=2(x+8)2−4
.
y=2x2+32x+124
(1 point)
Responses
The y-intercept is (0,124)
.
The y-intercept is left parenthesis 0 comma 124 right parenthesis .
The vertex of the quadratic is (−8,−4)
.
The vertex of the quadratic is left parenthesis negative 8 comma negative 4 right parenthesis .
The vertex of the quadratic is (8,−4)
.
The vertex of the quadratic is left parenthesis 8 comma negative 4 right parenthesis .
The quadratic function in vertex form is y=2(x+8)2−4
.
Answers
GPT-4o mini
To determine which statement is false about the quadratic function \( y = 2x^2 + 32x + 124 \), we can analyze it step by step.
1. **Finding the y-intercept**:
The y-intercept occurs when \( x = 0 \).
\[
y = 2(0)^2 + 32(0) + 124 = 124
\]
Thus, the y-intercept is \( (0, 124) \). This statement is **true**.
2. **Finding the vertex**:
The vertex \( (h, k) \) of a quadratic in standard form \( y = ax^2 + bx + c \) can be found using the formula:
\[
h = -\frac{b}{2a}
\]
Here, \( a = 2 \) and \( b = 32 \):
\[
h = -\frac{32}{2 \cdot 2} = -\frac{32}{4} = -8
\]
To find \( k \), substitute \( x = -8 \) back into the function:
\[
k = 2(-8)^2 + 32(-8) + 124 = 2(64) - 256 + 124 = 128 - 256 + 124 = -4
\]
Thus, the vertex is \( (-8, -4) \). This statement is **true**.
3. **Checking the vertex again**:
The statement "The vertex of the quadratic is \( (8, -4) \)" is incorrect, as we have determined that the vertex is \( (-8, -4) \). This statement is **false**.
4. **Finding the vertex form**:
The vertex form of a quadratic function can be derived from \( y = a(x - h)^2 + k \). We found \( h = -8 \) and \( k = -4 \), therefore:
\[
y = 2(x + 8)^2 - 4
\]
This confirms that the statement "The quadratic function in vertex form is \( y = 2(x + 8)^2 - 4 \)" is **true**.
In summary, the false statement is:
- "The vertex of the quadratic is \( (8, -4) \)."
1. **Finding the y-intercept**:
The y-intercept occurs when \( x = 0 \).
\[
y = 2(0)^2 + 32(0) + 124 = 124
\]
Thus, the y-intercept is \( (0, 124) \). This statement is **true**.
2. **Finding the vertex**:
The vertex \( (h, k) \) of a quadratic in standard form \( y = ax^2 + bx + c \) can be found using the formula:
\[
h = -\frac{b}{2a}
\]
Here, \( a = 2 \) and \( b = 32 \):
\[
h = -\frac{32}{2 \cdot 2} = -\frac{32}{4} = -8
\]
To find \( k \), substitute \( x = -8 \) back into the function:
\[
k = 2(-8)^2 + 32(-8) + 124 = 2(64) - 256 + 124 = 128 - 256 + 124 = -4
\]
Thus, the vertex is \( (-8, -4) \). This statement is **true**.
3. **Checking the vertex again**:
The statement "The vertex of the quadratic is \( (8, -4) \)" is incorrect, as we have determined that the vertex is \( (-8, -4) \). This statement is **false**.
4. **Finding the vertex form**:
The vertex form of a quadratic function can be derived from \( y = a(x - h)^2 + k \). We found \( h = -8 \) and \( k = -4 \), therefore:
\[
y = 2(x + 8)^2 - 4
\]
This confirms that the statement "The quadratic function in vertex form is \( y = 2(x + 8)^2 - 4 \)" is **true**.
In summary, the false statement is:
- "The vertex of the quadratic is \( (8, -4) \)."