Create a graph; then answer the following question.

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) (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)

Answer 1

To analyze the quadratic function \( y = x^2 - 8x \), we can rewrite it in vertex form to identify key points:

1. Find the vertex: The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \), \( b = -8 \), and \( c = 0 \).

   Calculate the x-coordinate of the vertex: \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

   Then, substitute \( x = 4 \) back into the original equation to find the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Thus, the vertex is at \( (4, -16) \).

2. Find the x-intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factor the equation: \[ 0 = x(x - 8) \] Setting each factor to zero gives: \[ x = 0 \quad \text{or} \quad x = 8 \] Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

3. Find the y-intercept: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 8 \cdot 0 = 0 \] Thus, the y-intercept is \( (0, 0) \).

Summarizing the key points:

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

Conclusion

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

Based on the choices provided:

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