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Complete the square to identify the vertex of y=x2−16x+14

.(1 point)
Responses

(8,−50)
left parenthesis 8 comma negative 50 right parenthesis

(8,78)
left parenthesis 8 comma 78 right parenthesis

(8,30)
left parenthesis 8 comma 30 right parenthesis

(8,14)

7 answers

To complete the square for the equation \( y = x^2 - 16x + 14 \) and find the vertex, follow these steps:

  1. Identify the quadratic and linear terms: The expression is \( x^2 - 16x \).

  2. Complete the square:

    • Take the coefficient of \( x \), which is \( -16 \), divide it by 2 to get \( -8 \), and then square it to get \( 64 \).
    • Rewrite \( x^2 - 16x \) as \( (x - 8)^2 - 64 \).

  3. Substitute back into the equation: \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

  4. Identify the vertex: The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the completed square form \( y = (x - 8)^2 - 50 \), we can see that:

    • \( h = 8 \)
    • \( k = -50 \)

Thus, the vertex is \( (8, -50) \).

Therefore, the correct response is: (8, -50).

Identify the vertex of y=−10(x+15)2−18
.(1 point)
Responses

(−10,−15)
left parenthesis negative 10 comma negative 15 right parenthesis

(−15,−18)
left parenthesis negative 15 comma negative 18 right parenthesis

(15,−18)
left parenthesis 15 comma negative 18 right parenthesis

(−150,−18)
left parenthesis negative 150 comma negative 18 right parenthesis

The equation given is in the vertex form of a parabola, which is represented as \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

The given equation is:

\[ y = -10(x + 15)^2 - 18 \]

To identify the vertex, we can rewrite the term \( (x + 15) \) in the form \( (x - h) \):

\[ y = -10(x - (-15))^2 - 18 \]

From the equation, we can see that:

  • \( h = -15 \)
  • \( k = -18 \)

Thus, the vertex of the parabola is:

(−15, −18).

Therefore, the correct response is: (−15, −18).

Create a graph; then answer the following 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-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)

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: (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

To analyze the function \( y = x^2 - 8x \) and find the key points (vertex, x-intercepts, and y-intercept), let's follow these steps:

  1. Vertex:

    • The standard form of the quadratic is \( y = ax^2 + bx + c \). Here, \( a = 1 \), \( b = -8 \), and \( c = 0 \).
    • The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \): \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
    • Substitute \( x = 4 \) back into the function to find the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]
    • 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 \( y = 0\): \[ 0 = x^2 - 8x \]
      • Factor the equation: \[ 0 = x(x - 8) \]
      • Setting each factor to zero gives: \[ x = 0 \quad \text{and} \quad x = 8 \]
    • Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

Summary of Key Points

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

Given the options, the correct response that matches our findings is:

  • x-intercepts: (0,0) and (8,0)
  • y-intercept: (0,0)
  • vertex: (4,−16)

Final Answer:

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

What point is symmetrical to (−1,−17)
for the graph of y=−x2−16
?(1 point)
Responses

(−1,17)
left parenthesis negative 1 comma 17 right parenthesis

(1,17)
left parenthesis 1 comma 17 right parenthesis

(1,−17)
left parenthesis 1 comma negative 17 right parenthesis

(−17,−1)

To find the point symmetrical to \((-1, -17)\) with respect to the graph of the equation \(y = -x^2 - 16\), we need to first identify the vertex of the parabola represented by this equation.

The standard form of a parabola is \(y = ax^2 + bx + c\). In this case, we can see:

\[ y = -x^2 - 16 \]

This is a downward-opening parabola, and it is in vertex form already with its vertex at:

  • The x-coordinate of the vertex for \(y = -x^2 - 16\) can be found by recognizing that the parabola is vertical and symmetric around the y-axis. Given that there’s no \(x\) term, the vertex is at \((0, -16)\).

Finding the Symmetrical Point

The symmetry axis of the parabola is the vertical line \(x = 0\) (the y-axis). To find the symmetrical point with respect to the y-axis, we take the original point \((-1, -17)\) and reflect it across this line.

  1. The x-coordinate of the symmetrical point will be the opposite sign of the original x-coordinate:

    • Original point: \(x = -1\)
    • Symmetrical point: \(x = 1\)

  2. The y-coordinate remains the same:

    • Original point: \(y = -17\)

Thus, the symmetrical point is:

\[ (1, -17) \]

Conclusion

Among the given options, the correct response is: (1, -17).