Question

Create a graph; then answer the following question.

What are the key points on the graph of y=x2−8x ? Name the vertex, x-intercept(s), and y-intercept.

(1 point)
Responses

x-intercepts: (0,0) (8,0)

y-intercept: (0,0)

vertex: (4,−16)

x -intercepts: left parenthesis 0 comma 0 right parenthesis left parenthesis 8 comma 0 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis

x-intercept: (4,−16)

y-intercept: (0,0)

vertex: (8,0)

x -intercept: left parenthesis 4 comma negative 16 right parenthesis y -intercept: left parenthesis 0 comma 0 right parenthesis vertex: left parenthesis 8 comma 0 right parenthesis

x-intercept: (0,8)

y-intercept:

Answer 1

To analyze the quadratic function \( y = x^2 - 8x \) and find the key points, we can start by rewriting the equation into vertex form or finding the relevant intercepts directly.

1. **Find the vertex:**
The vertex form of a parabola can be derived from the standard form \( y = ax^2 + bx + c \).
The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = -8 \).
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
Now plug \( x = 4 \) back into the original equation to find the y-coordinate:
\[
y = (4)^2 - 8 \cdot 4 = 16 - 32 = -16
\]
Thus, the vertex is \( (4, -16) \).

2. **Find the x-intercepts:**
To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factor the equation:
\[
0 = x(x - 8)
\]
This gives x-intercepts at \( x = 0 \) and \( x = 8 \), resulting in the points:
\[
(0, 0) \quad \text{and} \quad (8, 0)
\]

3. **Find the y-intercept:**
To find the y-intercept, set \( x = 0 \):
\[
y = 0^2 - 8 \cdot 0 = 0
\]
Thus, the y-intercept is \( (0, 0) \).

### Summary of Key Points:
- **Vertex:** \( (4, -16) \)
- **x-intercepts:** \( (0, 0) \) and \( (8, 0) \)
- **y-intercept:** \( (0, 0) \)

### Final Response:
The correct option is:
- x-intercepts: \( (0, 0), (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)