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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. (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-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-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)

1 answer

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

  1. Vertex:
    The vertex of a quadratic in the form \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
    \[ x = -\frac{b}{2a} \]
    In this case, \( a = 1 \) and \( b = -8 \), so:
    \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
    To find the y-coordinate, substitute \( x = 4 \) back into the equation:
    \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]
    Therefore, the vertex is \( (4, -16) \).

  2. X-Intercepts:
    To find the x-intercepts, set \( y = 0 \):
    \[ 0 = x^2 - 8x \]
    Factor the equation:
    \[ 0 = x(x - 8) \]
    Thus, the solutions are \( x = 0 \) and \( x = 8 \).
    Hence, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

  3. Y-Intercept:
    To find the y-intercept, set \( x = 0 \):
    \[ y = 0^2 - 8(0) = 0 \]
    Therefore, the y-intercept is \( (0, 0) \).

Summary of Key Points:

  • Vertex: \( (4, -16) \)
  • X-intercepts: \( (0, 0) \) and \( (8, 0) \)
  • Y-intercept: \( (0, 0) \)

Correct Response:

The correct responses corresponding to the analysis above are:

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

Now, looking at the options you provided, this analysis would correspond to:

  • x-intercepts: \( (0, 0) \) (not explicitly mentioned in any option, but included)
  • vertex: \( (4, -16) \)
  • y-intercept: \( (0, 0) \)

This matches the summary derived from the function.