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Create a graph; then answer the following question.

What are the key points on the graph of y=x2−8x
𝑦
=
𝑥
2
−
8
𝑥
? Name the vertex, x-intercept(s), and y-intercept.

(1 point)
Responses

x-intercept: (4,−16)
(
4
,
−
16
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (8,0)
(
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
,
0
)
(8,0)
(
8
,
0
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (4,−16)
(
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)
(
0
,
8
)

y-intercept: (0,0)
(
0
,
0
)

vertex: (4,−16)
(
4
,
−
16
)

x -intercep

1 answer

To analyze the quadratic function \( y = x^2 - 8x \), we can identify key points on its graph, including the vertex, x-intercepts, and y-intercept.

  1. Vertex: The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] For our function, \( a = 1 \) 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 function: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Therefore, the vertex is at: \[ (4, -16) \]

  2. X-intercepts: The x-intercepts occur where \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] Setting each factor to zero provides the solutions: \[ x = 0 \quad \text{or} \quad x = 8 \] Therefore, the x-intercepts are: \[ (0, 0) \quad \text{and} \quad (8, 0) \]

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

Summary of Key Points:

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

None of the response options entirely represent the summary correctly because:

  • The vertex is indeed \( (4, -16) \).
  • The x-intercepts are \( (0, 0) \) and \( (8, 0) \).
  • The y-intercept is also \( (0, 0) \).

Therefore, the correct response acknowledges the vertex as \( (4, -16) \) and lists the intercepts correctly:

  • Correct Answer from the options:
    • Vertex: \( (4, -16) \)
    • X-intercepts: \( (0, 0) \), \( (8, 0) \)
    • Y-intercept: \( (0, 0) \)