Question
Graph the equation y, equals, minus, x, squared, plus, 14, x, minus, 48y=−x
2
+14x−48 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the equation of the axis of symmetry.
Click to plot points. Click points to delete them.
Answer
Attempt 2 out of 2
Answer type:
Two numbers
one point
one equation
2
+14x−48 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the equation of the axis of symmetry.
Click to plot points. Click points to delete them.
Answer
Attempt 2 out of 2
Answer type:
Two numbers
one point
one equation
Answers
Answer
Find the equation of the axis of symmetry of the following parabola algebraically.
y, equals, minus, x, squared, plus, 8, x, minus, 20
y=−x
2
+8x−20
y, equals, minus, x, squared, plus, 8, x, minus, 20
y=−x
2
+8x−20
Answer
Find the equation of the axis of symmetry of the following parabola algebraically.
y, equals, minus, x, squared, minus, 2, x, plus, 7
y=−x
2
−2x+7
y, equals, minus, x, squared, minus, 2, x, plus, 7
y=−x
2
−2x+7
Answered by
GPT-4o mini
To graph the equation \( y = -x^2 + 14x - 48 \), we first need to find some important points, including the roots and the vertex.
### Step 1: Find the roots
To find the roots, we can set the equation equal to zero:
\[
-x^2 + 14x - 48 = 0
\]
Multiplying the entire equation by -1 to ease the calculations:
\[
x^2 - 14x + 48 = 0
\]
Next, we factor the quadratic:
\[
(x - 6)(x - 8) = 0
\]
Thus, the roots are:
\[
x = 6 \quad \text{and} \quad x = 8
\]
### Step 2: Find the vertex
The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = -1 \) and \( b = 14 \):
\[
x = -\frac{14}{2 \cdot -1} = 7
\]
Now, substitute \( x = 7 \) back into the equation to find the y-coordinate:
\[
y = -7^2 + 14 \cdot 7 - 48
\]
\[
y = -49 + 98 - 48 = 1
\]
Thus, the vertex is at the point \( (7, 1) \).
### Step 3: Additional points
We can choose some additional points around the vertex to get a better picture of the graph. Let's evaluate the function at \( x = 5 \) and \( x = 9 \):
1. **When \( x = 5 \)**:
\[
y = -(5)^2 + 14(5) - 48 = -25 + 70 - 48 = -3 \quad (5, -3)
\]
2. **When \( x = 9 \)**:
\[
y = -(9)^2 + 14(9) - 48 = -81 + 126 - 48 = -3 \quad (9, -3)
\]
### Summary of points
So the points to plot are:
1. Roots: \( (6, 0) \) and \( (8, 0) \)
2. Vertex: \( (7, 1) \)
3. Additional points: \( (5, -3) \) and \( (9, -3) \)
### Step 4: Axis of symmetry
The axis of symmetry can be found as the vertical line that passes through the x-coordinate of the vertex. Since the vertex is at \( (7, 1) \), the axis of symmetry is:
\[
x = 7
\]
### Final Answer
Now compiling everything together:
- Points to plot:
1. \( (6, 0) \)
2. \( (8, 0) \)
3. \( (7, 1) \)
4. \( (5, -3) \)
5. \( (9, -3) \)
- Axis of symmetry:
\[
x = 7
\]
So the final answer, formatted as requested:
- Plot points: \((6, 0)\), \((8, 0)\), \((7, 1)\), \((5, -3)\), and \((9, -3)\)
- Axis of symmetry: \(x = 7\)
### Step 1: Find the roots
To find the roots, we can set the equation equal to zero:
\[
-x^2 + 14x - 48 = 0
\]
Multiplying the entire equation by -1 to ease the calculations:
\[
x^2 - 14x + 48 = 0
\]
Next, we factor the quadratic:
\[
(x - 6)(x - 8) = 0
\]
Thus, the roots are:
\[
x = 6 \quad \text{and} \quad x = 8
\]
### Step 2: Find the vertex
The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = -1 \) and \( b = 14 \):
\[
x = -\frac{14}{2 \cdot -1} = 7
\]
Now, substitute \( x = 7 \) back into the equation to find the y-coordinate:
\[
y = -7^2 + 14 \cdot 7 - 48
\]
\[
y = -49 + 98 - 48 = 1
\]
Thus, the vertex is at the point \( (7, 1) \).
### Step 3: Additional points
We can choose some additional points around the vertex to get a better picture of the graph. Let's evaluate the function at \( x = 5 \) and \( x = 9 \):
1. **When \( x = 5 \)**:
\[
y = -(5)^2 + 14(5) - 48 = -25 + 70 - 48 = -3 \quad (5, -3)
\]
2. **When \( x = 9 \)**:
\[
y = -(9)^2 + 14(9) - 48 = -81 + 126 - 48 = -3 \quad (9, -3)
\]
### Summary of points
So the points to plot are:
1. Roots: \( (6, 0) \) and \( (8, 0) \)
2. Vertex: \( (7, 1) \)
3. Additional points: \( (5, -3) \) and \( (9, -3) \)
### Step 4: Axis of symmetry
The axis of symmetry can be found as the vertical line that passes through the x-coordinate of the vertex. Since the vertex is at \( (7, 1) \), the axis of symmetry is:
\[
x = 7
\]
### Final Answer
Now compiling everything together:
- Points to plot:
1. \( (6, 0) \)
2. \( (8, 0) \)
3. \( (7, 1) \)
4. \( (5, -3) \)
5. \( (9, -3) \)
- Axis of symmetry:
\[
x = 7
\]
So the final answer, formatted as requested:
- Plot points: \((6, 0)\), \((8, 0)\), \((7, 1)\), \((5, -3)\), and \((9, -3)\)
- Axis of symmetry: \(x = 7\)
Answered by
GPT-4o mini
To find the equation of the axis of symmetry for the parabola given by the equation
\[
y = -x^2 + 8x - 20,
\]
we can use the formula for the axis of symmetry for a parabola of the form \(y = ax^2 + bx + c\). The axis of symmetry can be found using the formula:
\[
x = -\frac{b}{2a},
\]
where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\) in the quadratic equation.
### Step 1: Identify \(a\) and \(b\)
In the given equation:
- \(a = -1\) (the coefficient of \(x^2\))
- \(b = 8\) (the coefficient of \(x\))
### Step 2: Apply the formula
Using the formula for the axis of symmetry:
\[
x = -\frac{b}{2a} = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4.
\]
### Conclusion
Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 + 8x - 20\) is:
\[
\boxed{x = 4}.
\]
\[
y = -x^2 + 8x - 20,
\]
we can use the formula for the axis of symmetry for a parabola of the form \(y = ax^2 + bx + c\). The axis of symmetry can be found using the formula:
\[
x = -\frac{b}{2a},
\]
where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\) in the quadratic equation.
### Step 1: Identify \(a\) and \(b\)
In the given equation:
- \(a = -1\) (the coefficient of \(x^2\))
- \(b = 8\) (the coefficient of \(x\))
### Step 2: Apply the formula
Using the formula for the axis of symmetry:
\[
x = -\frac{b}{2a} = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4.
\]
### Conclusion
Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 + 8x - 20\) is:
\[
\boxed{x = 4}.
\]
Answered by
GPT-4o mini
To find the equation of the axis of symmetry for the parabola given by the equation
\[
y = -x^2 - 2x + 7,
\]
we can use the formula for the axis of symmetry for a quadratic equation in the standard form \(y = ax^2 + bx + c\). The formula to find the axis of symmetry is:
\[
x = -\frac{b}{2a},
\]
where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).
### Step 1: Identify \(a\) and \(b\)
In the given equation:
- \(a = -1\) (the coefficient of \(x^2\))
- \(b = -2\) (the coefficient of \(x\))
### Step 2: Apply the formula
Now, we can substitute the values of \(a\) and \(b\) into the axis of symmetry formula:
\[
x = -\frac{b}{2a} = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1.
\]
### Conclusion
Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 - 2x + 7\) is:
\[
\boxed{x = -1}.
\]
\[
y = -x^2 - 2x + 7,
\]
we can use the formula for the axis of symmetry for a quadratic equation in the standard form \(y = ax^2 + bx + c\). The formula to find the axis of symmetry is:
\[
x = -\frac{b}{2a},
\]
where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).
### Step 1: Identify \(a\) and \(b\)
In the given equation:
- \(a = -1\) (the coefficient of \(x^2\))
- \(b = -2\) (the coefficient of \(x\))
### Step 2: Apply the formula
Now, we can substitute the values of \(a\) and \(b\) into the axis of symmetry formula:
\[
x = -\frac{b}{2a} = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1.
\]
### Conclusion
Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 - 2x + 7\) is:
\[
\boxed{x = -1}.
\]
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