Question

What are they key points on the graph of y = x^2 - 8x? Name the vertex, x intercept, and y intercept

Answer 1

To determine the key points of the quadratic function \( y = x^2 - 8x \), we can follow these steps:

1. **Vertex**: The vertex of a quadratic function 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 original equation to find the \( y \)-coordinate of the vertex:

\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]

Therefore, the vertex is at the point \( (4, -16) \).

2. **Y-intercept**: The y-intercept occurs when \( x = 0 \).

\[
y = (0)^2 - 8(0) = 0
\]

Thus, the y-intercept is at the point \( (0, 0) \).

3. **X-intercepts**: The x-intercepts occur when \( y = 0 \). We set the equation to zero and solve for \( x \):

\[
0 = x^2 - 8x
\]

Factoring gives:

\[
0 = x(x - 8)
\]

Setting each factor to zero results in:

\[
x = 0 \quad \text{or} \quad x = 8
\]

Thus, the x-intercepts are at the points \( (0, 0) \) and \( (8, 0) \).

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

Keep in mind that the y-intercept and one of the x-intercepts coincide at \( (0, 0) \).