Question

Decide if the following are true or false about the function

y = 3x^2 − 12x + 18

(4 points)
Vertex: (2,6)


Axis of Symmetry: x = 3


Maximum: 6


Y-intercept: 18

Answer 1

To determine whether the statements about the function \( y = 3x^2 − 12x + 18 \) are true or false, we will analyze the function step by step.

1. **Vertex**:
The vertex of a quadratic function in standard form \( y = ax^2 + bx + c \) can be determined using the formula for the x-coordinate:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 3 \) and \( b = -12 \):
\[
x = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2
\]
Now we substitute \( x = 2 \) back into the function to find the y-coordinate:
\[
y = 3(2)^2 − 12(2) + 18 = 3(4) - 24 + 18 = 12 - 24 + 18 = 6
\]
So, the vertex is \( (2, 6) \).
**True**.

2. **Axis of Symmetry**:
The axis of symmetry of a quadratic function is the vertical line that runs through the vertex. Since we found the vertex to be \( (2, 6) \), the axis of symmetry is \( x = 2 \).
The statement claims it's \( x = 3 \).
**False**.

3. **Maximum**:
Since the coefficient of \( x^2 \) (which is \( a = 3 \)) is positive, the parabola opens upwards, meaning it has a minimum point, not a maximum. The minimum value occurs at the vertex, which is \( y = 6 \).
Since there is no maximum value, the statement claiming the maximum is 6 is.
**False**.

4. **Y-intercept**:
The y-intercept occurs when \( x = 0 \). Let's substitute \( x = 0 \) into the function:
\[
y = 3(0)^2 − 12(0) + 18 = 18
\]
Therefore, the y-intercept is indeed 18.
**True**.

### Summary of Truth Values:
- Vertex: **True**
- Axis of Symmetry: **False**
- Maximum: **False**
- Y-intercept: **True**