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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: (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) (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.

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-intercept: (0,8)

y-intercept: (0,0)

vertex: (4,−16)

1 answer

To analyze the function \( y = x^2 - 8x \), we can rewrite it in standard form by completing the square.

  1. Rewrite the function: \[ y = x^2 - 8x \] Completing the square: \[ y = (x^2 - 8x + 16) - 16 \] \[ y = (x - 4)^2 - 16 \]

  2. Identify key points:

    • Vertex: The vertex of the parabola in the form \(y = (x - h)^2 + k\) is \((h, k)\). In this case, the vertex is: \[ (4, -16) \]

    • Y-Intercept: The y-intercept is found by setting \(x = 0\): \[ y = 0^2 - 8(0) = 0 \] So, the y-intercept is: \[ (0, 0) \]

    • X-Intercepts: The x-intercepts are found by setting \(y = 0\): \[ 0 = x^2 - 8x \] Factoring: \[ x(x - 8) = 0 \] Thus, the x-intercepts are: \[ x = 0 \quad \text{and} \quad x = 8 \] So the x-intercepts are: \[ (0, 0) \text{ and } (8, 0) \]

  3. Summary of key points:

    • Vertex: \((4, -16)\)
    • Y-Intercept: \((0, 0)\)
    • X-Intercepts: \((0, 0)\) and \((8, 0)\)

Based on this analysis, the correct responses for the key points on the graph of \(y = x^2 - 8x\) are:

  • Vertex: \( (4, -16) \)
  • X-Intercepts: \( (0, 0) \) and \( (8, 0) \)
  • Y-Intercept: \( (0, 0) \)

You can choose the response that correctly identifies these points. The correct option seems to be:

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