Question

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

Responses

There are no zeros.

(−1,0)(5,0)

(1,0)(−5,0)

(0,15)

Answer 1

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

Responses

(8,30)

(8,14)

(8,−50)

(8,78)

Answer 2

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

Responses

(15,−18)

(−150,−18)

(−15,−18)

(−10,−15)

Answer 3

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 4

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 5

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 6

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 7

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 8

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 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)**