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Use any method to locate the zeros of y=−3x^2+12x+15 .(1 point) Responses (0,15) left parenthesis 0 comma 15 right parenthesis There are no zeros. There are no zeros. (1,0)(−5,0) left parenthesis 1 comma 0 right parenthesis left parenthesis negative 5 comma 0 right parenthesis (−1,0)(5,0) left parenthesis negative 1 comma 0 right parenthesis left parenthesis 5 comma 0 right parenthesis

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

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

First, we can simplify the equation by dividing through by \(-3\):

\[ x^2 - 4x - 5 = 0 \]

Next, we can factor this quadratic equation. We look for two numbers that multiply to \(-5\) (the constant term) and add to \(-4\) (the coefficient of the linear term). These numbers are \(-5\) and \(1\). Therefore, we can factor the equation as follows:

\[ (x - 5)(x + 1) = 0 \]

Next, we set each factor to zero and solve for \( x \):

  1. \( x - 5 = 0 \) implies \( x = 5 \)
  2. \( x + 1 = 0 \) implies \( x = -1 \)

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

The corresponding points where the graph intersects the x-axis (the zeros) are:

  • \( (5, 0) \)
  • \( (-1, 0) \)

So the response contains two answers: \( (5, 0) \) and \( (-1, 0) \).

Complete the square to identify the vertex of y=x^2−16x+14 .(1 point) Responses (8,30) left parenthesis 8 comma 30 right parenthesis (8,−50) left parenthesis 8 comma negative 50 right parenthesis (8,14) left parenthesis 8 comma 14 right parenthesis (8,78)

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

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

  2. Complete the square for \( x^2 - 16x \):

    • Take half of the coefficient of \( x \) (which is \(-16\)), square it, and add/subtract it inside the parentheses.
    • Half of \(-16\) is \(-8\), and squaring it gives \( 64 \).

    So we rewrite the equation: \[ y = (x^2 - 16x + 64 - 64) + 14 \] This simplifies to: \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

  3. Identify the vertex: The vertex form of a quadratic \( y = a(x - h)^2 + k \) shows that the vertex is \((h, k)\). In our case, we have:

    • \( h = 8 \)
    • \( k = -50 \)

Thus, the vertex of the quadratic \( y = x^2 - 16x + 14 \) is at the point \( (8, -50) \).

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

Identify the vertex of y=−10(x+15)^2−18 .(1 point) Responses (15,−18) left parenthesis 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) left parenthesis negative 15 comma negative 18 right parenthesis

To identify the vertex of the function \( y = -10(x + 15)^2 - 18 \), we can observe it is already written in vertex form, which is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

From the given equation:

  • \( a = -10 \)
  • The expression inside the parentheses is \( (x + 15) \), which can be rewritten as \( (x - (-15)) \). This means \( h = -15 \).
  • The constant term is \( k = -18 \).

So, the vertex is at:

\[ (h, k) = (-15, -18) \]

Thus, the vertex of the given function is \( (-15, -18) \).

The correct response is \( (−15,−18) \).

Create a graph; then answer the following question. What are the key points on the graph of y=x^2−8x ? 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-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: (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.

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

1. Finding the Vertex

To find the vertex of the quadratic function in standard form \( y = ax^2 + bx + c \), we can use the formula for the x-coordinate of the vertex:

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

Here, \( a = 1 \) and \( b = -8 \):

\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

Now we substitute \( x = 4 \) back into the function to find the y-coordinate of the vertex:

\[ y = (4)^2 - 8(4) = 16 - 32 = -16 \]

So the vertex is at:

\[ (4, -16) \]

2. Finding the x-intercepts

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

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

Factoring the equation:

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

This gives us the solutions:

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

Thus, the x-intercepts are:

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

3. Finding the y-intercept

To find the y-intercept, we 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: \( (0, 0) \)

Based on this analysis, the correct response is:

  • x-intercepts: \( (0, 0) \) and \( (8, 0) \)
  • y-intercept: \( (0, 0) \)
  • vertex: \( (4, -16) \)

This matches the response option:

x-intercepts: \( (0, 0) \) and \( (8, 0) \); y-intercept: \( (0, 0) \); vertex: \( (4, -16) \).

What point is symmetrical to (−1,−17) for the graph of y=−x^2−16 ?(1 point) Responses (1,17) left parenthesis 1 comma 17 right parenthesis (−17,−1) left parenthesis negative 17 comma negative 1 right parenthesis (−1,17) left parenthesis negative 1 comma 17 right parenthesis (1,−17)

To find the point that is symmetrical to \((-1, -17)\) with respect to the graph of the equation \(y = -x^2 - 16\), we need to determine the axis of symmetry of the parabola defined by this equation.

Step 1: Identify the Vertex

The equation \(y = -x^2 - 16\) is in vertex form of a parabola, where the vertex is located at \((h, k)\) from the standard form \(y = a(x - h)^2 + k\).

For our equation:

  • The vertex occurs when \(x = 0\) because the \(x^2\) term has no linear component (the parabola opens downward).

Substituting \(x = 0\):

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

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

Step 2: Find the Symmetrical Point

The axis of symmetry is the vertical line \(x = 0\), which means that for any point \((x, y)\) on one side of the line, its symmetrical point will have the same \(y\)-coordinate but an opposite \(x\)-coordinate.

Given the point \((-1, -17)\):

  • The \(x\)-coordinate is \(-1\), which is \(1\) unit to the left of the axis of symmetry at \(x = 0\).
  • Thus, its symmetrical point will be \(1\) unit to the right of the axis of symmetry.

The symmetrical point will have the same \(y\)-coordinate:

\[ (1, -17) \]

Conclusion

The point that is symmetrical to \((-1, -17)\) with respect to the graph of \(y = -x^2 - 16\) is:

\[ (1, -17) \]

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