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-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: (0,0)

vertex: (4,−16)

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

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

vertex: (4,−16)

There is no y-intercept.

x -intercepts: left parenthesis 0 comma 0 right parenthesis left parenthesis 0 comma 8 right parenthesis vertex: left parenthesis 4 comma negative 16 right parenthesis There is no y -intercept.

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

Answer 1

To analyze the quadratic function \( y = x^2 - 8x \), we can rewrite it in vertex form by completing the square.

1. **Complete the square**:
\[
y = x^2 - 8x
\]
To complete the square:
\[
y = (x^2 - 8x + 16) - 16 = (x - 4)^2 - 16
\]

Thus, the vertex form of the equation is:
\[
y = (x - 4)^2 - 16
\]

From this form, we can identify key points:
- **Vertex**: The vertex is located at \( (4, -16) \).

2. **Finding the x-intercepts**:
To find the x-intercepts, set \( y = 0 \):
\[
0 = (x - 4)^2 - 16
\]
\[
(x - 4)^2 = 16
\]
Taking the square root:
\[
x - 4 = 4 \quad \text{or} \quad x - 4 = -4
\]
Therefore:
\[
x = 8 \quad \text{and} \quad x = 0
\]
The x-intercepts are:
- \( (0, 0) \)
- \( (8, 0) \)

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

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

### Answer Selection:
Given the options provided, the correct response is:
- **x-intercepts**: \( (0, 0) \) and \( (8, 0) \)
- **y-intercept**: \( (0, 0) \)
- **vertex**: \( (4, -16) \)

So, the correct answer aligns with:
x-intercepts: \( (0, 0) \) and \( (8, 0) \), y-intercept: \( (0, 0) \), vertex: \( (4, -16) \).