Question

Create a graph; then answer the following question. What are the key points on the graph of y = x^2 - 8x? ? Name the vertex, x-intercept(s), and y-intercept.

Answer 1

To analyze the graph of the quadratic function \( y = x^2 - 8x \), we can rewrite it in vertex form and identify key points such as the vertex, x-intercepts, and y-intercept.

### Step 1: Find the vertex
The vertex form of a quadratic is given by:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. To find the vertex, we can complete the square for the quadratic equation \( y = x^2 - 8x \).

1. Start with the quadratic:
\[
y = x^2 - 8x
\]

2. To complete the square, take half of the coefficient of \(x\) (which is -8), square it, and then add and subtract it inside the equation:
\[
y = (x^2 - 8x + 16) - 16
\]
Here, \( ( -4 )^2 = 16 \).

3. Now, rewrite the equation:
\[
y = (x - 4)^2 - 16
\]

From this, we see that the vertex \((h, k)\) is:
- Vertex: \((4, -16)\)

### Step 2: Find the x-intercepts
To find the x-intercepts, set \(y = 0\) and solve for \(x\):
\[
0 = x^2 - 8x
\]

Factor the equation:
\[
0 = x(x - 8)
\]

Setting each factor to zero gives:
- \(x = 0\)
- \(x = 8\)

Thus, the x-intercepts are:
- X-intercepts: \((0, 0)\) and \((8, 0)\)

### Step 3: Find the y-intercept
To find the y-intercept, set \(x = 0\) in the original equation:
\[
y = 0^2 - 8(0) = 0
\]

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

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

### Graph Representation
To visualize this in a graph (please imagine a parabola):
- The vertex is at \((4, -16)\), which is the lowest point of the parabola.
- The parabola opens upwards because the coefficient of \(x^2\) is positive.
- The x-intercepts at \((0, 0)\) and \((8, 0)\) indicate where the graph crosses the x-axis.
- The y-intercept is also at \((0, 0)\), showing that the graph intersects the y-axis at the origin.

If you would like a graphical representation, you can use graphing software or graphing calculators to plot these points and the curve.