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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: (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

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) (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-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.
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1 answer

To analyze the quadratic function \( y = x^2 - 8x \), we can complete the square or use the vertex form to find the key points.

Step 1: Find the Vertex

The function can be expressed in vertex form by completing the square:

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

The vertex of the parabola is at the point \( (4, -16) \).

Step 2: Find the x-intercepts

The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] Thus, the x-intercepts are \( x = 0 \) and \( x = 8 \), corresponding to the points:

  • \( (0, 0) \)
  • \( (8, 0) \)

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: \( (0, 0) \)

Response Selections

Based on the analysis, the correct key points are:

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

So the most accurate response would be:

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