Question

Use any method to locate the zeros of y=−3x2+12x+15.(1 point)
Responses

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

There are no zeros.
There are no zeros.

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

(1,0)(−5,0)

Answer 1

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)
left parenthesis 8 comma 14 right parenthesis

Answer 2

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

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

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

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

(15,−18)

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.

(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.

Answer 4

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

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

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

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

(−17,−1)

Answer 5

To find the zeros of the quadratic function \( y = -3x^2 + 12x + 15 \), we can set \( y \) to 0 and solve for \( x \):

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

Next, we can rearrange the equation:

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

Now, we can apply the quadratic formula, where \( a = 3 \), \( b = -12 \), and \( c = -15 \):

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

Calculating the discriminant \( b^2 - 4ac \):

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

Now applying the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{324}}{2 \times 3}
\]
\[
x = \frac{12 \pm 18}{6}
\]

This gives us two solutions:

1. \( x = \frac{12 + 18}{6} = \frac{30}{6} = 5 \)
2. \( x = \frac{12 - 18}{6} = \frac{-6}{6} = -1 \)

So, the zeros of the function are \( (5, 0) \) and \( (-1, 0) \).

The correct response is:
\((-1, 0)(5, 0)\)

Answer 6

To complete the square for the quadratic function \( y = x^2 - 16x + 14 \), we follow these steps:

1. **Identify the coefficient of \( x \)**: Here, the coefficient of \( x \) is \(-16\).

2. **Take half of this coefficient and square it**:
\[
\left(\frac{-16}{2}\right)^2 = (-8)^2 = 64
\]

3. **Add and subtract this square inside the equation**:
\[
y = (x^2 - 16x + 64) + 14 - 64
\]
\[
y = (x - 8)^2 - 50
\]

Now, we can identify the vertex of the parabola from the completed square equation \( y = (x - 8)^2 - 50 \).

The vertex is at the point \( (h, k) \), where \( h = 8 \) and \( k = -50 \).

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

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

Answer 7

The given equation is in the vertex form of a quadratic function, which is expressed as:

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

Here, \( (h, k) \) represents the vertex of the parabola.

In your equation:

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

we can rewrite the expression to match the vertex form:

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

From this, we can identify that:

- \( h = -15 \)
- \( k = -18 \)

Thus, the vertex of the function is:

\[
(-15, -18)
\]

The correct response is:
\((-15, -18)\)

Answer 8

To analyze the quadratic function \( y = x^2 - 8x \), we can identify its key points: the vertex, x-intercepts, and y-intercept.

### Step 1: Find the Vertex

The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
h = -\frac{b}{2a}
\]

In this case:
- \( a = 1 \)
- \( b = -8 \)

Calculating \( h \):
\[
h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]

Now, substitute \( h \) back into the original equation to find \( k \):
\[
k = (4)^2 - 8(4) = 16 - 32 = -16
\]

So, the vertex is \( (4, -16) \).

### Step 2: Find the x-intercepts

To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 8x
\]
Factoring gives:
\[
0 = x(x - 8)
\]
Thus, the x-intercepts are:
\[
x = 0 \quad \text{and} \quad x = 8
\]

So, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).

### Step 3: Find the y-intercept

To find 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**: \( (0, 0) \)

### Correct Response

The correct response is:
- **x-intercepts**: \( (0, 0) \) and \( (8, 0) \)
- **y-intercept**: \( (0, 0) \)
- **vertex**: \( (4, -16) \)

So, the overall correct answer is:
**x-intercepts: \( (0, 0) \) \( (8, 0) \); y-intercept: \( (0, 0) \); vertex: \( (4, -16) \)**.

Answer 9

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

### Step 1: Identify the vertex of the parabola

The graph of the function \(y = -x^2 - 16\) is a downward-opening parabola. The vertex form of a parabola \(y = a(x - h)^2 + k\) shows that the vertex is located at \((h, k)\).

Here, for \(y = -x^2 - 16\):
- The vertex is at \((0, -16)\).

### Step 2: Find the line of symmetry

For a parabola given by \(y = ax^2 + bx + c\), the line of symmetry is vertical and can be expressed as \(x = h\), where \(h\) is the x-coordinate of the vertex. Here, the line of symmetry is at \(x = 0\).

### Step 3: Find the symmetrical point

To find the symmetrical point with respect to \(x = 0\), we need to reflect the point \((-1, -17)\). The reflection across the line \(x = 0\) changes the x-coordinate, while the y-coordinate remains the same.

- The x-coordinate of the point \((-1, -17)\) reflects to \(1\):
- Original point: \(x = -1\)
- Reflected point: \(x = 1\)

The y-coordinate remains \(-17\).

### Conclusion

The symmetrical point to \((-1, -17)\) is \((1, -17)\).

The correct response is:
\((1, -17)\)