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

Question

.(1 point)
Responses

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

(8,14)
left parenthesis 8 comma 14 right parenthesis

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

(8,30)

Answer 1

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

1. Identify the coefficients: The equation is in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -16 \), and \( c = 14 \).

2. Complete the square:

   • Start with the quadratic and linear terms: \( x^2 - 16x \).
   • To complete the square, take half of the coefficient of \( x \) (which is \(-16\)), square it, and add and subtract it inside the equation. Half of \(-16\) is \(-8\) and squaring it gives \( 64 \).

   Thus, \[ y = (x^2 - 16x + 64) - 64 + 14 \]

3. Rewrite the equation:

   • Now the equation becomes: \[ y = (x - 8)^2 - 50 \]

4. Identify the vertex:

   • The vertex form of the equation is \( y = a(x - h)^2 + k \), where the vertex is \((h, k)\).
   • From \( y = (x - 8)^2 - 50 \), we see that \( h = 8 \) and \( k = -50 \).

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

The correct response is: (8, -50).

Answer 2

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

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

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

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

(−10,−15)

Answer 3

The equation is already in vertex form:

\[ y = a(x - h)^2 + k \]

where \((h, k)\) represents the vertex.

For the equation:

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

we can rewrite it in the standard vertex form:

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

From this, you can see that \(h = -15\) and \(k = -18\). Therefore, the vertex of the quadratic function is:

\[ (-15, -18). \]

The correct response is: (−15, −18).

Answer 4

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-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)
(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-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.
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Answer 5

To find the key points on the graph of the quadratic function \( y = x^2 - 8x \), we will:

1. Find the vertex by completing the square.
2. Find the x-intercepts by setting \( y = 0 \).
3. Find the y-intercept by evaluating \( y \) when \( x = 0 \).

1. Finding the Vertex

The equation is in standard form \( y = ax^2 + bx \). To complete the square:

\[ y = x^2 - 8x \]

Take half of the coefficient of \( x \) (which is -8), square it, and add and subtract it:

\[ y = (x^2 - 8x + 16) - 16 \]

Rewrite:

\[ y = (x - 4)^2 - 16 \]

Thus, the vertex is at:

\[ (4, -16) \]

2. Finding the x-Intercepts

Set \( y = 0 \):

\[ 0 = x^2 - 8x \] \[ 0 = x(x - 8) \]

This gives us \( x = 0 \) and \( x = 8 \), so the x-intercepts are:

\[ (0, 0) \quad \text{and} \quad (8, 0) \]

3. Finding the y-Intercept

Set \( x = 0 \):

\[ y = 0^2 - 8(0) = 0 \]

So, the y-intercept is:

\[ (0, 0) \]

Summary of Key Points

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

Conclusion

The correct response is:

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

So, the option that corresponds with these values is: 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.

Answer 6

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

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

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

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

(−1,17)

Answer 7

To find the point that is symmetrical to \((-1, -17)\) on the graph of the function \(y = -x^2 - 16\), we must first identify the axis of symmetry of the parabola.

1. Determine the axis of symmetry: The equation \(y = -x^2 - 16\) is a downward-opening parabola. The vertex form of a parabola \(y = a(x - h)^2 + k\) shows that the axis of symmetry is the vertical line \(x = h\).

For the equation:

• The parabola is centered at \(x = 0\) (the line of symmetry is \(x = 0\) when \(y = -x^2 - 16\) is rewritten).

2. Finding the symmetrical point: Since the axis of symmetry is \(x = 0\), the distance from the point \((-1, -17)\) to the axis of symmetry (the y-axis) is \(1\) unit to the left (because \(-1\) is negative). The symmetrical point will be \(1\) unit to the right of the y-axis, which is \(1\).

3. Keep the y-coordinate the same: The y-coordinate remains unchanged for the symmetrical point because we reflect it over the axis of symmetry. Therefore, the y-coordinate remains \(-17\).

Thus, the symmetrical point to \((-1, -17)\) across the axis of symmetry \(x = 0\) is \((1, -17)\).

Conclusion

The correct response is: (1, −17).