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-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

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

Answer 1

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

### Step 1: Completing the Square
To complete the square for \( y = x^2 - 8x \):

1. Take half of the coefficient of \( x \) (which is -8), square it, and add it inside the equation:
- Half of -8 is -4, and squaring it gives 16.

2. Rewrite the equation:
\[
y = (x^2 - 8x + 16) - 16
\]
\[
y = (x - 4)^2 - 16
\]

This shows that the vertex of the parabola is at \( (4, -16) \).

### Step 2: Finding the Intercepts

#### Vertex:
- The vertex is at \((4, -16)\).

#### x-intercepts:
To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factoring gives:
\[
0 = x(x - 8)
\]
This gives the solutions:
- \( x = 0 \) and \( x = 8 \)
So the x-intercepts are:
- \((0, 0)\) and \((8, 0)\).

#### y-intercept:
To find the y-intercept, set \( x = 0 \):
\[
y = 0^2 - 8(0) = 0
\]
So the y-intercept is:
- \((0, 0)\).

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

### Answering the Original Question
Putting this all together, here are the key points again:
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)

Therefore, the correct response includes:
- x-intercepts: \( (0, 0) \) (and \( (8, 0) \))
- vertex: \( (4, -16) \)
- y-intercept: \( (0, 0) \)