Question
Decide if the following are true or false about the function y=3x2−12x+18y=3x2−12x+18 (4 points) True False Vertex: (2,6) Axis of Symmetry: x = 3 Maximum: 6 Y-intercept: 18
Answers
GPT-4o mini
Let's analyze the function \( y = 3x^2 - 12x + 18 \) to verify the statements given.
### 1. **Vertex: (2, 6)**
The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the vertex formula \( x = -\frac{b}{2a} \).
Here, \( a = 3 \) and \( b = -12 \):
\[
x = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2
\]
Now plug \( x = 2 \) back into the function to find \( y \):
\[
y = 3(2)^2 - 12(2) + 18 = 3(4) - 24 + 18 = 12 - 24 + 18 = 6
\]
So, the vertex is indeed \( (2, 6) \).
**True**
### 2. **Axis of Symmetry: x = 3**
The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Since the vertex we found is \( (2, 6) \), the axis of symmetry is \( x = 2 \), not \( x = 3 \).
**False**
### 3. **Maximum: 6**
A parabola opens upwards when \( a > 0 \) (in this case, \( a = 3 > 0 \)), indicating that this function has a minimum value, not a maximum. The minimum value occurs at the vertex \( (2, 6) \).
**False**
### 4. **Y-intercept: 18**
The y-intercept occurs when \( x = 0 \):
\[
y = 3(0)^2 - 12(0) + 18 = 18
\]
Thus, the y-intercept is 18.
**True**
### Summary:
- Vertex: (2, 6) — **True**
- Axis of Symmetry: x = 3 — **False**
- Maximum: 6 — **False**
- Y-intercept: 18 — **True**
Final answers:
- Vertex: True
- Axis of Symmetry: False
- Maximum: False
- Y-intercept: True
### 1. **Vertex: (2, 6)**
The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the vertex formula \( x = -\frac{b}{2a} \).
Here, \( a = 3 \) and \( b = -12 \):
\[
x = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2
\]
Now plug \( x = 2 \) back into the function to find \( y \):
\[
y = 3(2)^2 - 12(2) + 18 = 3(4) - 24 + 18 = 12 - 24 + 18 = 6
\]
So, the vertex is indeed \( (2, 6) \).
**True**
### 2. **Axis of Symmetry: x = 3**
The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Since the vertex we found is \( (2, 6) \), the axis of symmetry is \( x = 2 \), not \( x = 3 \).
**False**
### 3. **Maximum: 6**
A parabola opens upwards when \( a > 0 \) (in this case, \( a = 3 > 0 \)), indicating that this function has a minimum value, not a maximum. The minimum value occurs at the vertex \( (2, 6) \).
**False**
### 4. **Y-intercept: 18**
The y-intercept occurs when \( x = 0 \):
\[
y = 3(0)^2 - 12(0) + 18 = 18
\]
Thus, the y-intercept is 18.
**True**
### Summary:
- Vertex: (2, 6) — **True**
- Axis of Symmetry: x = 3 — **False**
- Maximum: 6 — **False**
- Y-intercept: 18 — **True**
Final answers:
- Vertex: True
- Axis of Symmetry: False
- Maximum: False
- Y-intercept: True