Sure, let's analyze the given function step-by-step:
### Problem:
We're given the function [tex]\( y = x^2 - 8x \)[/tex].
### Step-by-Step Analysis:
1. Identifying the Function Type:
- The given function [tex]\( y = x^2 - 8x \)[/tex] is a quadratic function. Quadratic functions are generally of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- Comparing, we see [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Finding the Vertex:
- The vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex].
[tex]\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\][/tex]
- To find the y-coordinate of the vertex, substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[
y = 4^2 - 8(4) = 16 - 32 = -16
\][/tex]
- Therefore, the vertex of the function is at [tex]\( (4, -16) \)[/tex].
3. Finding the Axis of Symmetry:
- The axis of symmetry of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
[tex]\[
x = 4
\][/tex]
4. Finding the x-intercepts:
- The x-intercepts are the points where the graph of the function crosses the x-axis.
- Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = x^2 - 8x
\][/tex]
- Factor the quadratic expression:
[tex]\[
0 = x(x - 8)
\][/tex]
- Solve each factor:
[tex]\[
x = 0 \quad \text{or} \quad x - 8 = 0 \Rightarrow x = 8
\][/tex]
- Therefore, the x-intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex], which are the points [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex].
5. Finding the y-intercept:
- The y-intercept is the point where the graph of the function crosses the y-axis.
- Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[
y = 0^2 - 8(0) = 0
\][/tex]
- Therefore, the y-intercept is at [tex]\( (0, 0) \)[/tex].
### Summary:
1. Function Type: Quadratic function [tex]\( y = x^2 - 8x \)[/tex]
2. Vertex: [tex]\( (4, -16) \)[/tex]
3. Axis of Symmetry: [tex]\( x = 4 \)[/tex]
4. x-intercepts: [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex]
5. y-intercept: [tex]\( (0, 0) \)[/tex]
The graph of the given function is a parabola that opens upwards with its vertex at [tex]\( (4, -16) \)[/tex]. The parabola intersects the x-axis at [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex], and intersects the y-axis at [tex]\( (0, 0) \)[/tex].
