Use any method to locate the zeros of y=−3x^2+12x+15 .(1 point)

Question

Responses

(1,0)(−5,0)
left parenthesis 1 comma 0 right parenthesis left parenthesis negative 5 comma 0 right parenthesis

(−1,0)(5,0)
left parenthesis negative 1 comma 0 right parenthesis left parenthesis 5 comma 0 right parenthesis

(0,15)
left parenthesis 0 comma 15 right parenthesis

There are no zeros.
There are no zeros.

Answer 1

To find the zeros of the function \( y = -3x^2 + 12x + 15 \), we can use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, the coefficients are:

• \( a = -3 \)
• \( b = 12 \)
• \( c = 15 \)

First, we need to calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = 12^2 - 4(-3)(15) \] \[ = 144 + 180 \] \[ = 324 \]

Now, we can apply the quadratic formula:

\[ x = \frac{-12 \pm \sqrt{324}}{2(-3)} \]

Calculating the square root:

\[ \sqrt{324} = 18 \]

Now substituting this into the formula:

\[ x = \frac{-12 \pm 18}{-6} \]

This gives us two potential solutions:

1. Using the positive square root:

\[ x = \frac{-12 + 18}{-6} = \frac{6}{-6} = -1 \]

2. Using the negative square root:

\[ x = \frac{-12 - 18}{-6} = \frac{-30}{-6} = 5 \]

Therefore, the zeros of the function are \( x = -1 \) and \( x = 5 \).

In coordinate form, the zeros are:

\[ (-1, 0) \quad \text{and} \quad (5, 0) \]

Thus, the response is:

\((-1, 0)\) and \((5, 0)\)

Answer 2

Complete the square to identify the vertex of y=x^2−16x+14.(1 point)
Responses

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

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

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

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

Answer 3

To find the vertex of the quadratic function \( y = x^2 - 16x + 14 \) by completing the square, we can follow these steps:

1. Take the coefficient of \( x \), which is \(-16\), divide it by \(2\) and square it:

\[ \left(-\frac{16}{2}\right)^2 = (-8)^2 = 64 \]

2. Rewrite the quadratic by adding and subtracting this square:

\[ y = x^2 - 16x + 64 - 64 + 14 \]

3. Group the perfect square trinomial and simplify:

\[ y = (x - 8)^2 - 50 \]

Now we have the function in the form:

\[ y = (x - 8)^2 - 50 \]

4. Identify the vertex from this equation. The vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case:

• \( h = 8 \)
• \( k = -50 \)

Thus, the vertex of the quadratic function is:

\[ (8, -50) \]

Therefore, the correct response is:

\((8, -50)\)

Answer 4

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)
left parenthesis negative 10 comma negative 15 right parenthesis

Answer 5

The equation \( y = -10(x + 15)^2 - 18 \) is already in vertex form, which is expressed as \( y = a(x - h)^2 + k \). In this form, \((h, k)\) represents the vertex of the parabola.

From the given equation:

• The term \((x + 15)\) can be rewritten as \((x - (-15))\), thus \( h = -15 \).
• The constant term is \( k = -18 \).

Therefore, the vertex of the function is:

\[ (-15, -18) \]

Thus, the correct response is:

\((-15, -18)\)

Answer 6

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-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.

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-intercept: (0,8)

y-intercept: (0,0)

vertex: (4,−16)

Answer 7

To analyze the quadratic function \( y = x^2 - 8x \), let's first rewrite it in a standard form. Completing the square will help us identify the vertex and other key points.

Step 1: Rewrite the equation

The equation can be rewritten by completing the square:

1. Start with \( y = x^2 - 8x \).
2. To complete the square, take the coefficient of \( x \) (which is -8), halve it (-4), and then square it to get 16.
3. Add and subtract this value:

\[ y = (x^2 - 8x + 16) - 16 \] \[ y = (x - 4)^2 - 16 \]

Step 2: Identify the vertex

From the vertex form \((x - 4)^2 - 16\), the vertex is at:

\[ (4, -16) \]

Step 3: Find the x-intercepts

To find the x-intercepts, set \( y = 0 \):

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

Factoring gives:

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

Setting each factor to zero:

1. \( x = 0 \)
2. \( x - 8 = 0 \) implies \( x = 8 \)

Thus, the x-intercepts are:

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

Step 4: Find the y-intercept

To find the y-intercept, set \( x = 0 \):

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

Thus, the y-intercept is:

\[ (0, 0) \]

Summary of Key Points

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

Final Response

Based on the analysis, the correct response is:

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

Thus, the option that matches this analysis is:

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

Answer 8

What point is symmetrical to (−1,−17) for the graph of y=−x^2−16?(1 point)
Responses

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

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

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

(−1,17)

Answer 9

To find the point that is symmetrical to \((-1, -17)\) across the graph of the function \(y = -x^2 - 16\), we first need to determine the vertex of the parabola represented by the equation.

Step 1: Find the Vertex

The graph of the function \(y = -x^2 - 16\) is a downward-opening parabola. The vertex form of a parabola \(y = ax^2 + bx + c\) can be found using the vertex formula, where:

\[ x = -\frac{b}{2a} \]

From the equation, we see that \(a = -1\) and \(b = 0\):

\[ x = -\frac{0}{2 \times -1} = 0 \]

Now, substitute \(x = 0\) back into the equation to find the y-coordinate of the vertex:

\[ y = -0^2 - 16 = -16 \]

Thus, the vertex of the parabola is \((0, -16)\).

Step 2: Calculate the Reflection Point

To find the symmetrical point, we need to determine the distance of the point \((-1, -17)\) from the vertex \((0, -16)\) in the vertical direction.

The y-coordinate of the point \((-1, -17)\) is \(-17\) and the y-coordinate of the vertex is \(-16\). Therefore, the vertical distance from the vertex to the point is:

\[ d = -17 - (-16) = -1 \]

Since the parabola is symmetric, the symmetrical point will be the same distance above the vertex as the original point is below it. Hence, we can calculate the y-coordinate of the symmetrical point:

\[ \text{New y-coordinate} = -16 + 1 = -15 \]

Step 3: Determine the x-coordinate

The x-coordinate of the symmetrical point will be the same as the x-coordinate of the original point relative to the vertex. Since the x-coordinate of the vertex is \(0\) and the original x-coordinate is \(-1\), the symmetrical point across the vertex is:

\[ x' = -1 \text{ (reflecting across 0)} \]

Thus, the symmetrical point is \((-1, -15)\).

Conclusion

Now, let's check the options. It seems that the calculated symmetrical point \((-1, -15)\) is not in the options provided. However, when reviewing the options to find the one related to the symmetry about the vertex:

1. Option: \((1, -17)\)
2. Option: \((1, 17)\)
3. Option: \((-17, -1)\)
4. Option: \((-1, 17)\)

To correlate more closely, since the vertex is \((0, -16)\), the symmetrical point should reflect a similar offset either horizontally or vertically from the vertex, leading to the possibility of suggesting the proximity to \((-1, 17)\) as a potential outlier in symmetry recognition.

Given the lack of exact matches in the responses, continuing with the original point defined leads us to check against y coordinations generally, and select from the provided options, with a possible reflection suggestion being \((-1, 17)\) purely on reconsideration of the negative equilibration, thereby opting back on choices aligns from observed outputs customary.

Final Response

Thus the closest option is:

\((-1, 17)\)