Question

Decide if the following are true or false about the function below.
y=3x^2−12x+18


(4 points)
True False
Vertex: (2,6)


Axis of Symmetry: x = 3


Maximum: 6


y-intercept: 18

Answer 1

Let's analyze the function \( y = 3x^2 - 12x + 18 \) step by step to determine the truth of each statement:

1. **Vertex: (2, 6)**

To find the vertex of the quadratic function \( y = ax^2 + bx + c \), we use the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
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 the y-coordinate:
\[
y = 3(2)^2 - 12(2) + 18 = 3(4) - 24 + 18 = 12 - 24 + 18 = 6
\]
Therefore, the vertex is indeed \( (2, 6) \).
**True**

2. **Axis of Symmetry: x = 3**

The axis of symmetry for a parabola in the form \( y = ax^2 + bx + c \) is given by the same x-value we calculated for the vertex:
\[
x = 2
\]
So the axis of symmetry is \( x = 2 \), not \( x = 3 \).
**False**

3. **Maximum: 6**

Since the coefficient \( a \) (which is 3) is positive, the parabola opens upwards, which means it has a minimum point (the vertex) rather than a maximum. Thus, the vertex's y-coordinate, 6, is a minimum value, not a maximum.
**False**

4. **y-intercept: 18**

The y-intercept occurs where \( x = 0 \):
\[
y = 3(0)^2 - 12(0) + 18 = 18
\]
Therefore, the y-intercept is indeed 18.
**True**

In summary:
- Vertex: (2, 6) - **True**
- Axis of Symmetry: x = 3 - **False**
- Maximum: 6 - **False**
- y-intercept: 18 - **True**