Question

Create a graph; then answer the following question.

Given the quadratic equation y=x^2−8x
, name the vertex, x-intercepts (s), and y-intercept.
Desmos Graphing Calculator

(1 point)
Responses

x-intercept: (0, 8)

y-intercept: (0, 0)

vertex: (4, -16)

x-intercept: (0, 8) y-intercept: (0, 0) vertex: (4, -16)

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

vertex: (4, 16)

There is no y-intercept.

x-intercepts: (0, 0) and (0, 8) vertex: (4, 16) There is no y-intercept.

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

y-intercept: (0, 0)

vertex: (4, -16)

x-intercepts: (0, 0) and (8, 0) y-intercept: (0, 0) vertex : (4, -16)

x-intercept: (4, -16)

y-intercept: (0, 0)

vertex: (8, 0)

Answer 1

To analyze the given quadratic equation \( y = x^2 - 8x \), we can follow these steps:

1. **Find the vertex**:
The vertex of a parabola represented by \( y = ax^2 + bx + c \) 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 \( y \):
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]
Therefore, the vertex is \( (4, -16) \).

2. **Find the x-intercepts**:
The x-intercepts occur where \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factoring gives:
\[
0 = x(x - 8)
\]
Setting each factor equal to zero gives us:
\[
x = 0 \quad \text{or} \quad x - 8 = 0 \implies x = 8
\]
Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

3. **Find the y-intercept**:
The y-intercept occurs where \( x = 0 \):
\[
y = (0)^2 - 8(0) = 0
\]
Therefore, the y-intercept is \( (0, 0) \).

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

Given the choices presented, the correct response is:
**x-intercepts: (0, 0) and (8, 0); y-intercept: (0, 0); vertex: (4, -16)**.