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

Question

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

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 2

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 3

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 4

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 5

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 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.

(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 7

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 8

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 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: \((1, -17)\)