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)

3 answers

To analyze the function \( y = x^2 - 8x \), we can follow these steps:

  1. Find the vertex: The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \), where \((h,k)\) is the vertex. To find the vertex of the quadratic in standard form \( y = ax^2 + bx + c \), we can use the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] In our case:

    • \( a = 1 \)
    • \( b = -8 \)

    Plugging in the values: \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

    To find the y-coordinate of the vertex, substitute \( x = 4 \) back into the equation: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] So, the vertex is \( (4, -16) \).

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

  3. Find the x-intercepts: The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring out \( x \): \[ 0 = x(x - 8) \] This gives us two solutions: \[ x = 0 \quad \text{and} \quad x = 8 \] Thus, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

Now, summarizing the key points:

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

With this information, we can now respond directly to your question.

The correct key points are:

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

Therefore, the statement "x-intercept: (0,8) y-intercept: (0,0) vertex: (4,−16)" is incorrect for the x-intercept and correct for the vertex and y-intercept based on our calculations.

The accurate response should be:

  • X-intercepts: \( (0, 0) \) and \( (8, 0) \)
  • Y-intercept: \( (0, 0) \)
  • Vertex: \( (4, -16) \)
which one is it ?

Here are the correct key points for the graph of the function \( y = x^2 - 8x \):

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

So the summary of the main points is:

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

The options you provided earlier contain some inaccuracies. If we were to select from the provided options, the correct and complete point would include the vertex \( (4, -16) \) while correctly identifying that the x-intercepts include \( (0, 0) \) and \( (8, 0) \).

Let me know if you need additional clarification!