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

Question

Responses

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

(−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.

(1,0)(−5,0)

Answer 1

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

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

We can either factor the quadratic or use the quadratic formula. Let's start with the quadratic formula, which is given by:

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

For our equation, the coefficients are:

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

Substituting these values into the formula:

1. Calculate the discriminant \( b^2 - 4ac \):

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

2. Since the discriminant is positive, we have two distinct real roots. Now we calculate \( x \):

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

Now we can simplify it for both values of \( x \):

• For the positive root: \[ x = \frac{-12 + 18}{-6} = \frac{6}{-6} = -1 \]

• For the negative root: \[ x = \frac{-12 - 18}{-6} = \frac{-30}{-6} = 5 \]

Thus, the zeros of the function are \( x = -1 \) and \( x = 5 \). Therefore, we can express the zeros as points on the graph:

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

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

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

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

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

(8,78)

Answer 3

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

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

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

3. Complete the square:

   • Take the coefficient of \(x\) (-16), divide it by 2, and square it: \[ \left(\frac{-16}{2}\right)^2 = (-8)^2 = 64 \]
   • Add and subtract this value inside the parentheses: \[ y = (x^2 - 16x + 64 - 64) + 14 \] \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

4. Identify the vertex: The vertex form of a quadratic is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. From the equation \(y = (x - 8)^2 - 50\), we see that:

   • \(h = 8\)
   • \(k = -50\)

Thus, the vertex of the 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

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

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

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

(−150,−18)

Answer 5

The given equation is in vertex form:

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

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

From the equation:

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

Thus, the vertex \((h, k)\) 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.

(1 point)
Responses

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) (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-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-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

Answer 7

To identify the key points on the graph of the function \( y = x^2 - 8x \), we need to determine the vertex, x-intercepts, and y-intercept.

1. Identifying the Vertex:
   We can rewrite the equation in vertex form by completing the square.

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

   To complete the square, we take half of the coefficient of \(x\) (which is \(-8\)), square it, and adjust the equation:

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

   Therefore, we have the vertex at: \[ (4, -16) \]

2. Finding the X-Intercepts:
   To find the x-intercepts, we set \(y = 0\):

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

   Factoring this gives:

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

   Thus, the x-intercepts are:

   \[ x = 0 \quad \text{and} \quad x = 8 \]

   The corresponding points are \((0, 0)\) and \((8, 0)\).

3. Finding the Y-Intercept:
   The y-intercept occurs when \(x = 0\):

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

   Thus, the y-intercept is:

   \[ (0, 0) \]

Summary of Key Points:

• Vertex: \((4, -16)\)
• X-Intercepts: \((0, 0)\) and \((8, 0)\)
• Y-Intercept: \((0, 0)\)

Choosing the Correct Response:

The responses that match these findings should be combined. The correct response from your options appears to be:

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