Use any method to locate the zeros of y=−3x^2+12x+15

Question

Responses

There are no zeros.

(−1,0)(5,0)

(1,0)(−5,0)

(0,15)

Answer 1

To find the zeros of the quadratic function \( y = -3x^2 + 12x + 15 \), we need to set \( y \) equal to zero and solve for \( x \):

\[ -3x^2 + 12x + 15 = 0 \]

We can use the quadratic formula, which is given by:

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

where \( a = -3 \), \( b = 12 \), and \( c = 15 \).

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

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

Since the discriminant is positive, there will be two real roots.

Now we calculate the roots using the quadratic formula:

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

First, find \( \sqrt{324} = 18 \):

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

Now calculate the two possible values for \( x \):

1. \( x = \frac{-12 + 18}{-6} = \frac{6}{-6} = -1 \)

2. \( x = \frac{-12 - 18}{-6} = \frac{-30}{-6} = 5 \)

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

The correct response among the options provided is:

\((−1,0)(5,0)\)

Answer 2

Complete the square to identify the vertex of y=x^2−16x+14

Responses

(8,30)

(8,14)

(8,−50)

(8,78)

Answer 3

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

1. Start with the original function: \[ y = x^2 - 16x + 14 \]

2. Group the quadratic and linear terms: \[ y = (x^2 - 16x) + 14 \]

3. Complete the square for the expression inside the parentheses:

   • Take half of the coefficient of \( x \) (which is \(-16\)), square it, and add and subtract that value inside the parentheses.

   Half of \(-16\) is \(-8\), and \((-8)^2 = 64\):

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

   This simplifies to:

   \[ y = ((x - 8)^2 - 64) + 14 \]

4. Combine the constants: \[ y = (x - 8)^2 - 50 \]

5. Write the function in vertex form: \[ y = (x - 8)^2 - 50 \]

The vertex of the parabola is given by the point \( (h, k) \) from the vertex form \( y = a(x - h)^2 + k \).

Here, \( h = 8 \) and \( k = -50 \).

Thus, the vertex is: \[ (8, -50) \]

The correct response is: (8,−50)

Answer 4

Identify the vertex of y=−10(x+15)^2−18

Responses

(15,−18)

(−150,−18)

(−15,−18)

(−10,−15)

Answer 5

The vertex of a quadratic function in vertex form is given by the expression \( y = a(x - h)^2 + k \), where the vertex is at the point \( (h, k) \).

The given function is:

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

We can rewrite it to identify \( h \) and \( k \):

1. Notice that \((x + 15)^2\) can be interpreted as \((x - (-15))^2\). Thus, \(h = -15\).
2. The constant term \(-18\) is \(k\).

So, the vertex \((h, k)\) of the function \( y = -10(x + 15)^2 - 18 \) is:

\[ (-15, -18) \]

Therefore, 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=x2−8x
Name the vertex, x-intercept(s), and y-intercept.

Responses

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

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

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

x-intercepts: (0,0)(0,8),vertex: (4,−16),There is no y-intercept.

Answer 7

To analyze the function \( y = x^2 - 8x \), we will find the vertex, x-intercept(s), and y-intercept.

1. Finding the Vertex

We can express the function in vertex form by completing the square.

Starting with:

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

To complete the square:

1. Take half of the coefficient of \( x \) (which is -8), then square it: \[ \left(\frac{-8}{2}\right)^2 = 16 \]

2. Add and subtract this value inside the equation: \[ y = (x^2 - 8x + 16 - 16) \] \[ y = (x - 4)^2 - 16 \]

Thus, the vertex is:

\[ (4, -16) \]

2. Finding the x-intercepts

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

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

Factoring gives:

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

This provides the intercepts \( x = 0 \) and \( x = 8 \). Thus, the x-intercepts are:

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

3. Finding 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: \( (0, 0) \)

Conclusion:

Thus, the correct response 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
Responses:

(1,−17)

(−17,−1)

(1,17)

(−1,17)

Answer 9

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

Step 1: Identify the Vertex

The given function \( y = -x^2 - 16 \) is already in standard form.

The vertex of the parabola defined by \( y = ax^2 + bx + c \) occurs at:

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

In this case, \( a = -1 \) and \( b = 0 \):

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

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

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

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

Step 2: Find the Symmetrical Point

The point \( (-1, -17) \) is to the left of the vertex \( (0, -16) \). The distance from the vertex to the point is \( 1 \) unit to the left (from \( x = 0 \) to \( x = -1 \)).

To find the symmetrical point, we move the same distance to the right of the vertex:

\[ x = 0 + 1 = 1 \]

The y-coordinate does not change for symmetry across the vertex, so the y-coordinate remains:

\[ y = -17 \]

Thus, the symmetrical point is:

\[ (1, -17) \]

Conclusion

The correct response is:

(1,−17)