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: (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-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

Answer 1

To analyze the quadratic function \( y = x^2 - 8x \), let's follow these steps:

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

2. **Find the x-intercepts**:
Set \( y = 0 \) to find the x-intercepts:
\[
0 = x^2 - 8x
\]
Factor the equation:
\[
0 = x(x - 8)
\]
This gives us the solutions:
\[
x = 0 \quad \text{and} \quad x = 8
\]
Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

3. **Find the y-intercept**:
Set \( x = 0 \) to find the y-intercept:
\[
y = (0)^2 - 8(0) = 0
\]
This gives us the y-intercept \( (0, 0) \).

### Summary of Results
- **Vertex**: \( (4, -16) \)
- **X-intercepts**: \( (0, 0) \), \( (8, 0) \)
- **Y-intercept**: \( (0, 0) \)

### Correct Response
Considering the options you provided, the correct observations are:
- X-intercepts: \( (0, 0) \), \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
- Vertex: \( (4, -16) \)

Therefore, the correct choice from your list is:
- **x -intercepts**: \( \text{(0, 0), (8, 0)} \)
- **y -intercept**: \( \text{(0, 0)} \)
- **vertex**: \( \text{(4, -16)} \)