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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: (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) (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 find the key points on the graph of the quadratic function \( y = x^2 - 8x \), let's identify the vertex, x-intercepts, and y-intercept step-by-step.

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

  2. Y-Intercept: The y-intercept occurs when \( x = 0 \): \[ y = (0)^2 - 8(0) = 0 \] So, the y-intercept is \( (0, 0) \).

  3. X-Intercepts: The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring the equation: \[ 0 = x(x - 8) \] This gives us \( x = 0 \) and \( x = 8 \). Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

Summary of the key points:

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

Now, let's evaluate the provided options based on this information:

  1. Vertex: (4, -16) (correct)
  2. X-Intercepts: (0, 0) and (8, 0) (correct)
  3. Y-Intercept: (0, 0) (correct)

Thus, the accurate response is:

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