Sure, let's determine the vertex, axis of symmetry, and y-intercept for the quadratic function [tex]\( f(x) = 2x^2 - 12x + 24 \)[/tex].
### Step 1: Finding the Axis of Symmetry
For a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex], the axis of symmetry is given by the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
In this case, [tex]\( a = 2 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 24 \)[/tex]. Plugging in the values, we get:
[tex]\[
x = -\frac{-12}{2 \cdot 2} = \frac{12}{4} = 3.0
\][/tex]
So, the axis of symmetry is [tex]\( x = 3.0 \)[/tex].
### Step 2: Finding the Vertex
The vertex of a quadratic function is a point [tex]\((x, y)\)[/tex] where [tex]\(x\)[/tex] is the axis of symmetry and [tex]\(y\)[/tex] is the value of the function at that [tex]\(x\)[/tex]-coordinate.
We already have the [tex]\(x\)[/tex]-coordinate of the vertex, which is 3.0. To find the corresponding [tex]\(y\)[/tex]-coordinate, we substitute [tex]\( x = 3.0 \)[/tex] back into the original function:
[tex]\[
f(3.0) = 2(3.0)^2 - 12(3.0) + 24
\][/tex]
Calculating inside:
[tex]\[
f(3.0) = 2 \cdot 9 - 36 + 24 = 18 - 36 + 24 = 6.0
\][/tex]
So, the vertex is at [tex]\( (3.0, 6.0) \)[/tex].
### Step 3: Finding the Y-intercept
The y-intercept is the point where the graph crosses the y-axis, which happens when [tex]\( x = 0 \)[/tex].
To find the y-intercept, substitute [tex]\( x = 0 \)[/tex] into the original function:
[tex]\[
f(0) = 2(0)^2 - 12(0) + 24 = 0 + 0 + 24 = 24
\][/tex]
So, the y-intercept is [tex]\( 24 \)[/tex].
### Summary
The quadratic function [tex]\( f(x) = 2x^2 - 12x + 24 \)[/tex] has the following characteristics:
- Axis of Symmetry: [tex]\( x = 3.0 \)[/tex]
- Vertex: [tex]\( (3.0, 6.0) \)[/tex]
- Y-intercept: [tex]\( 24 \)[/tex]
These are the key features of the given quadratic function.
