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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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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)
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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
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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.
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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)
Answered by
GPT-4o mini
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 \( (-1, 0) \).
\[
-3x^2 + 12x + 15 = 0
\]
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 \( (-1, 0) \).
Answered by
GPT-4o mini
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 \( (8, -50) \).
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 \( (8, -50) \).
Answered by
GPT-4o mini
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 \( (−15,−18) \).
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 \( (−15,−18) \).
Answered by
GPT-4o mini
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:** \( (4, -16) \).
### 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:** \( (4, -16) \).
Answered by
GPT-4o mini
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 \( (1, -17) \).
### 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 \( (1, -17) \).