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.

Answer 1

To analyze the quadratic function \( y = x^2 - 8x \) and identify key points, we will find the vertex, x-intercepts, and y-intercept.

### Step 1: Find the Vertex
The vertex of a quadratic function in the standard 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 we can substitute \( x = 4 \) back into the function to find the y-coordinate of the vertex:

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

Thus, the vertex is:
\[
(4, -16)
\]

### Step 2: Find the x-intercepts
To find the x-intercepts, set \( y = 0 \) and solve for \( x \):

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

Factoring the equation gives:

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

Setting each factor to zero provides the x-intercepts:

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

Thus, the x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (8, 0)
\]

### Step 3: Find the y-intercept
The y-intercept occurs when \( x = 0 \):

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

Thus, the y-intercept is:
\[
(0, 0)
\]

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

### Graph
Though I cannot create visual graphs directly, you can plot the points as follows:
1. Draw the coordinate axes.
2. Plot the vertex at \( (4, -16) \).
3. Plot the x-intercepts at \( (0, 0) \) and \( (8, 0) \).
4. Notice that the y-intercept is the same as one of the x-intercepts, \( (0, 0) \).

The shape of the parabola opens upwards, and it passes through the points you have identified.