Question
Use any method to locate the zeros of y=−3x2+12x+15 .(1 point)
Responses
(0,15)
left parenthesis 0 comma 15 right parenthesis
(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
There are no zeros.
Responses
(0,15)
left parenthesis 0 comma 15 right parenthesis
(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
There are no zeros.
Answers
Answer
Complete the square to identify the vertex of y=x2−16x+14 .(1 point)
Responses
(8,14)
left parenthesis 8 comma 14 right parenthesis
(8,78)
left parenthesis 8 comma 78 right parenthesis
(8,−50)
left parenthesis 8 comma negative 50 right parenthesis
(8,30)
Responses
(8,14)
left parenthesis 8 comma 14 right parenthesis
(8,78)
left parenthesis 8 comma 78 right parenthesis
(8,−50)
left parenthesis 8 comma negative 50 right parenthesis
(8,30)
Answer
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)
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
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-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: (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-intercept: (0,8)
y-intercept: (0,0)
vertex: (4,−16)
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-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: (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-intercept: (0,8)
y-intercept: (0,0)
vertex: (4,−16)
Answer
What point is symmetrical to (−1,−17) for the graph of y=−x2−16 ?(1 point)
Responses
(−17,−1)
left parenthesis negative 17 comma negative 1 right parenthesis
(1,−17)
left parenthesis 1 comma negative 17 right parenthesis
(1,17)
left parenthesis 1 comma 17 right parenthesis
(−1,17)
Responses
(−17,−1)
left parenthesis negative 17 comma negative 1 right parenthesis
(1,−17)
left parenthesis 1 comma negative 17 right parenthesis
(1,17)
left parenthesis 1 comma 17 right parenthesis
(−1,17)
Answered by
GPT-4o mini
To find the zeros of the function \( y = -3x^2 + 12x + 15 \), we can set \( y = 0 \) and solve for \( x \):
\[
-3x^2 + 12x + 15 = 0
\]
We can simplify this equation by dividing all terms by \(-3\):
\[
x^2 - 4x - 5 = 0
\]
Next, we can factor the quadratic:
\[
(x - 5)(x + 1) = 0
\]
Setting each factor equal to zero gives us the potential zeros:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, the zeros of the function are \( x = -1 \) and \( x = 5 \).
Hence, the points where the function intersects the x-axis are:
\[
(-1, 0) \text{ and } (5, 0)
\]
The answer is:
\(\left( -1, 0 \right) \text{ and } \left( 5, 0 \right)\)
\[
-3x^2 + 12x + 15 = 0
\]
We can simplify this equation by dividing all terms by \(-3\):
\[
x^2 - 4x - 5 = 0
\]
Next, we can factor the quadratic:
\[
(x - 5)(x + 1) = 0
\]
Setting each factor equal to zero gives us the potential zeros:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, the zeros of the function are \( x = -1 \) and \( x = 5 \).
Hence, the points where the function intersects the x-axis are:
\[
(-1, 0) \text{ and } (5, 0)
\]
The answer is:
\(\left( -1, 0 \right) \text{ and } \left( 5, 0 \right)\)
Answered by
GPT-4o mini
To complete the square for the quadratic function \(y = x^2 - 16x + 14\) and identify the vertex, we follow these steps:
1. Start with the quadratic in standard form:
\[
y = x^2 - 16x + 14
\]
2. To complete the square, focus on the \(x^2 - 16x\) part. Take half of the coefficient of \(x\) (which is \(-16\)), square it, and add and subtract this value inside the equation:
- Half of \(-16\) is \(-8\).
- Squaring \(-8\) gives \(64\).
3. Rewrite the equation:
\[
y = (x^2 - 16x + 64) - 64 + 14
\]
4. This can be simplified to:
\[
y = (x - 8)^2 - 50
\]
Now, we can identify the vertex of the parabola from the equation \(y = (x - 8)^2 - 50\). The vertex form is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
From the completed square:
- \(h = 8\)
- \(k = -50\)
Thus, the vertex of the quadratic function occurs at the point \((8, -50)\).
Therefore, the correct response is:
\((8, -50)\)
1. Start with the quadratic in standard form:
\[
y = x^2 - 16x + 14
\]
2. To complete the square, focus on the \(x^2 - 16x\) part. Take half of the coefficient of \(x\) (which is \(-16\)), square it, and add and subtract this value inside the equation:
- Half of \(-16\) is \(-8\).
- Squaring \(-8\) gives \(64\).
3. Rewrite the equation:
\[
y = (x^2 - 16x + 64) - 64 + 14
\]
4. This can be simplified to:
\[
y = (x - 8)^2 - 50
\]
Now, we can identify the vertex of the parabola from the equation \(y = (x - 8)^2 - 50\). The vertex form is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
From the completed square:
- \(h = 8\)
- \(k = -50\)
Thus, the vertex of the quadratic function occurs at the point \((8, -50)\).
Therefore, the correct response is:
\((8, -50)\)
Answered by
GPT-4o mini
The given function is in vertex form:
\[
y = -10(x + 15)^2 - 18
\]
In the vertex form of a parabola, \(y = a(x - h)^2 + k\), the vertex is represented by the point \((h, k)\).
From the function \(y = -10(x + 15)^2 - 18\):
- The expression \((x + 15)\) can be rewritten as \((x - (-15))\), which means \(h = -15\).
- The value of \(k\) is \(-18\).
Thus, the vertex of the function is located at the point:
\[
(-15, -18)
\]
Therefore, the correct response is:
\((-15, -18)\)
\[
y = -10(x + 15)^2 - 18
\]
In the vertex form of a parabola, \(y = a(x - h)^2 + k\), the vertex is represented by the point \((h, k)\).
From the function \(y = -10(x + 15)^2 - 18\):
- The expression \((x + 15)\) can be rewritten as \((x - (-15))\), which means \(h = -15\).
- The value of \(k\) is \(-18\).
Thus, the vertex of the function is located at the point:
\[
(-15, -18)
\]
Therefore, the correct response is:
\((-15, -18)\)
Answered by
GPT-4o mini
To analyze the quadratic function \(y = x^2 - 8x\), we can determine the key points, specifically the vertex, x-intercepts, and y-intercept.
### Step 1: Find the Vertex
The vertex of a parabola given in the form \(y = ax^2 + bx + c\) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For \(y = x^2 - 8x\), we have:
- \(a = 1\)
- \(b = -8\)
Calculating the x-coordinate of the vertex:
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
To find the y-coordinate, we substitute \(x = 4\) back into the equation:
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]
Thus, 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:
\[
0 = x(x - 8)
\]
Setting each factor to zero gives the solutions:
\[
x = 0 \quad \text{and} \quad x = 8
\]
Therefore, the x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (8, 0)
\]
### Step 3: Find the y-intercept
To find the y-intercept, 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:** \((0, 0)\)
Given these findings, the correct response is:
**x-intercepts:** \((0,0)\) \((8,0)\)
**y-intercept:** \((0,0)\)
**vertex:** \((4,-16)\)
### Step 1: Find the Vertex
The vertex of a parabola given in the form \(y = ax^2 + bx + c\) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For \(y = x^2 - 8x\), we have:
- \(a = 1\)
- \(b = -8\)
Calculating the x-coordinate of the vertex:
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
To find the y-coordinate, we substitute \(x = 4\) back into the equation:
\[
y = (4)^2 - 8(4) = 16 - 32 = -16
\]
Thus, 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:
\[
0 = x(x - 8)
\]
Setting each factor to zero gives the solutions:
\[
x = 0 \quad \text{and} \quad x = 8
\]
Therefore, the x-intercepts are:
\[
(0, 0) \quad \text{and} \quad (8, 0)
\]
### Step 3: Find the y-intercept
To find the y-intercept, 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:** \((0, 0)\)
Given these findings, the correct response is:
**x-intercepts:** \((0,0)\) \((8,0)\)
**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 function \(y = -x^2 - 16\), we first need to identify the vertical line of symmetry for the parabola.
### Step 1: Determine the vertex of the parabola
The vertex form of a quadratic equation in the form \(y = ax^2 + bx + c\) tells us that the vertex can be found using the formula \(x = -\frac{b}{2a}\).
The given equation is:
\[
y = -x^2 - 16
\]
This is in the standard form where:
- \(a = -1\)
- \(b = 0\) (there is no \(x\) term)
The vertex's x-coordinate is:
\[
x = -\frac{0}{2 \cdot -1}= 0
\]
Substituting \(x = 0\) into the equation to find the y-coordinate:
\[
y = -0^2 - 16 = -16
\]
Thus, the vertex of the parabola is:
\[
(0, -16)
\]
### Step 2: Determine the symmetry
The line of symmetry for the parabola \(y = -x^2 - 16\) is the vertical line \(x = 0\) (the y-axis). Therefore, the point \((-1, -17)\) will have a symmetrical point that is the same distance from the y-axis but on the opposite side.
### Step 3: Find the symmetrical point
Since the x-coordinate of \((-1, -17)\) is \(-1\), its symmetrical x-coordinate will be \(1\) (which is the positive counterpart).
Maintaining the same y-coordinate (which is \(-17\)), the symmetrical point is:
\[
(1, -17)
\]
### Conclusion
Therefore, the answer is:
\((1, -17)\)
### Step 1: Determine the vertex of the parabola
The vertex form of a quadratic equation in the form \(y = ax^2 + bx + c\) tells us that the vertex can be found using the formula \(x = -\frac{b}{2a}\).
The given equation is:
\[
y = -x^2 - 16
\]
This is in the standard form where:
- \(a = -1\)
- \(b = 0\) (there is no \(x\) term)
The vertex's x-coordinate is:
\[
x = -\frac{0}{2 \cdot -1}= 0
\]
Substituting \(x = 0\) into the equation to find the y-coordinate:
\[
y = -0^2 - 16 = -16
\]
Thus, the vertex of the parabola is:
\[
(0, -16)
\]
### Step 2: Determine the symmetry
The line of symmetry for the parabola \(y = -x^2 - 16\) is the vertical line \(x = 0\) (the y-axis). Therefore, the point \((-1, -17)\) will have a symmetrical point that is the same distance from the y-axis but on the opposite side.
### Step 3: Find the symmetrical point
Since the x-coordinate of \((-1, -17)\) is \(-1\), its symmetrical x-coordinate will be \(1\) (which is the positive counterpart).
Maintaining the same y-coordinate (which is \(-17\)), the symmetrical point is:
\[
(1, -17)
\]
### Conclusion
Therefore, the answer is:
\((1, -17)\)