Question
calculate the radius of a circle by completing the square of the equation x^2 + y^2 - 16x - 10y + 40 = 0
a. -7
b. 4
c. 7
d. 6
Derive the equation of a parabola given the focus is at (-4,6) and the directris is ar y = 8. Fill in the missing values of the equation in standard form.
y = - 1/4 x^2 - __ x + 3
a. 2
b. -2
c. 1
d. -1
You are asked to graph a parabola on a coordinate plane given the equation y = - 1/8 x^2 + 3/4 x + 7/8. Fill in the missing value on the table for the coordinates of the parabola.
x-value
-1
3
___
11
y-value
0
2
0
-6
a. 7
b. -7
c. 6
d. -6
A certain statue at a momument has a perimeter of rope hung by poles, Between each pole the rope forms a U-shape like that, like a parabola, can be expresses by the equation y = (x = 3/2)^2 + 5/4. Suppose you were to graph the equation; at what point would the graph cross the y-axis?
y = ____
a. 7
b. -7
c. 3.5
d. -3.5
A golfer uses a tracking device to determine the data of the ball as it is in the air. She calculates that her ball, when 34 yards away from her, reached a maximum height of 72 yards before descending. The path of the golf ball can be modeled by a quadratic function where x represents the horizontal distance and y represents the height of the ball. In terms of the context, which key feature would represent the maximum height of the ball?
Option #1: vertex
Option #2: domain
Option #3: intercepts
Option #__ describes the point at which the ball reaches maximum height before descending.
A metronome is a device used to maintain a steady beat in music. The device has the shape of a pyramid.
Identify the number of sides of the two-dimensional vertical cross section.
The two-dimensional vertical cross section has __ sides
a. 2
b. 3
c. 4
d. 5
Which of the following shapes can be rotated to create a three-dimensional object like an orange?
Option #1: right triangle
Option #2: square
Option #3: hemisphere
Option #4: rectangle
Option #___
a. 1
b. 2
c. 3
d. 4
a. -7
b. 4
c. 7
d. 6
Derive the equation of a parabola given the focus is at (-4,6) and the directris is ar y = 8. Fill in the missing values of the equation in standard form.
y = - 1/4 x^2 - __ x + 3
a. 2
b. -2
c. 1
d. -1
You are asked to graph a parabola on a coordinate plane given the equation y = - 1/8 x^2 + 3/4 x + 7/8. Fill in the missing value on the table for the coordinates of the parabola.
x-value
-1
3
___
11
y-value
0
2
0
-6
a. 7
b. -7
c. 6
d. -6
A certain statue at a momument has a perimeter of rope hung by poles, Between each pole the rope forms a U-shape like that, like a parabola, can be expresses by the equation y = (x = 3/2)^2 + 5/4. Suppose you were to graph the equation; at what point would the graph cross the y-axis?
y = ____
a. 7
b. -7
c. 3.5
d. -3.5
A golfer uses a tracking device to determine the data of the ball as it is in the air. She calculates that her ball, when 34 yards away from her, reached a maximum height of 72 yards before descending. The path of the golf ball can be modeled by a quadratic function where x represents the horizontal distance and y represents the height of the ball. In terms of the context, which key feature would represent the maximum height of the ball?
Option #1: vertex
Option #2: domain
Option #3: intercepts
Option #__ describes the point at which the ball reaches maximum height before descending.
A metronome is a device used to maintain a steady beat in music. The device has the shape of a pyramid.
Identify the number of sides of the two-dimensional vertical cross section.
The two-dimensional vertical cross section has __ sides
a. 2
b. 3
c. 4
d. 5
Which of the following shapes can be rotated to create a three-dimensional object like an orange?
Option #1: right triangle
Option #2: square
Option #3: hemisphere
Option #4: rectangle
Option #___
a. 1
b. 2
c. 3
d. 4
Answers
Answered by
GPT-4o mini
Let's go through the questions one by one.
### 1. Calculate the radius of a circle by completing the square of the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\).
To complete the square, we rearrange the equation:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
Completing the square for \(x\) and \(y\):
1. For \(x^2 - 16x\):
- Take half of -16 (which is -8), square it (64), and add/subtract it inside the equation.
- So, \(x^2 - 16x + 64\).
2. For \(y^2 - 10y\):
- Take half of -10 (which is -5), square it (25), and add/subtract it inside the equation.
- So, \(y^2 - 10y + 25\).
Now, adding these gives:
\[
(x - 8)^2 + (y - 5)^2 - 64 - 25 + 40 = 0
\]
This simplifies to:
\[
(x - 8)^2 + (y - 5)^2 = 49
\]
Here, we have the equation of a circle centered at (8, 5) with a radius of \(\sqrt{49} = 7\).
**The answer is: c. 7**
---
### 2. Derive the equation of a parabola given the focus at (-4,6) and the directrix at \(y = 8\).
The vertex of the parabola is midway between the focus and the directrix. The distance from the focus to the directrix is 2 units (from 6 to 8), so the vertex is at:
\[
\left(-4, \frac{6 + 8}{2}\right) = (-4, 7)
\]
Since the focus is below the directrix, the parabola opens downwards. The distance from the vertex to the focus is 1. Thus, \(p = 1\).
The standard form of the equation is:
\[
(y - k) = -\frac{1}{4p}(x - h)^2
\]
Substituting \(h = -4\), \(k = 7\), and \(p = 1\):
\[
y - 7 = -\frac{1}{4}(x + 4)^2
\]
We can rearrange this to standard form \(y = -\frac{1}{4}(x + 4)^2 + 7\).
To match with \(y = -\frac{1}{4} x^2 - bx + 3\), we rewrite:
Taking the completed square form and equating terms, we need to fill in \(b\) where \(-b = 8\) from the standard form derived.
Thus, \(b = -2\).
**The answer is: b. -2**
---
### 3. Graph a parabola \(y = -\frac{1}{8} x^2 + \frac{3}{4} x + \frac{7}{8}\).
Let's find the missing x-value corresponding to the y-value of 0.
Set the equation to zero:
\[
0 = -\frac{1}{8} x^2 + \frac{3}{4} x + \frac{7}{8}
\]
To eliminate the fraction, multiply through by 8:
\[
0 = -x^2 + 6x + 7
\]
Rearranging gives:
\[
x^2 - 6x - 7 = 0
\]
Factoring:
\[
(x - 7)(x + 1) = 0
\]
Thus \(x = 7\) or \(x = -1\).
The missing x-value between -1 and 3 is \(7\).
**The answer is: a. 7**
---
### 4. The y-intercept of the parabola \(y = (x - \frac{3}{2})^2 + \frac{5}{4}\).
To find the y-intercept, set \(x = 0\):
\[
y = (0 - \frac{3}{2})^2 + \frac{5}{4} = \frac{9}{4} + \frac{5}{4} = \frac{14}{4} = \frac{7}{2} = 3.5
\]
**The answer is: c. 3.5**
---
### 5. The maximum height of the golf ball represented in the context.
The maximum height is represented by the **vertex**, which is the highest point of the quadratic function.
**The answer is: Option #1: vertex**
---
### 6. Identify the number of sides of the two-dimensional vertical cross-section of a pyramid.
The two-dimensional vertical cross-section of a pyramid has **3 sides** (forming a triangle).
**The answer is: b. 3**
---
### 7. Shape that can be rotated to create a three-dimensional object like an orange.
A **hemisphere** can be rotated to create a three-dimensional object like an orange.
**The answer is: c. 3**
### 1. Calculate the radius of a circle by completing the square of the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\).
To complete the square, we rearrange the equation:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
Completing the square for \(x\) and \(y\):
1. For \(x^2 - 16x\):
- Take half of -16 (which is -8), square it (64), and add/subtract it inside the equation.
- So, \(x^2 - 16x + 64\).
2. For \(y^2 - 10y\):
- Take half of -10 (which is -5), square it (25), and add/subtract it inside the equation.
- So, \(y^2 - 10y + 25\).
Now, adding these gives:
\[
(x - 8)^2 + (y - 5)^2 - 64 - 25 + 40 = 0
\]
This simplifies to:
\[
(x - 8)^2 + (y - 5)^2 = 49
\]
Here, we have the equation of a circle centered at (8, 5) with a radius of \(\sqrt{49} = 7\).
**The answer is: c. 7**
---
### 2. Derive the equation of a parabola given the focus at (-4,6) and the directrix at \(y = 8\).
The vertex of the parabola is midway between the focus and the directrix. The distance from the focus to the directrix is 2 units (from 6 to 8), so the vertex is at:
\[
\left(-4, \frac{6 + 8}{2}\right) = (-4, 7)
\]
Since the focus is below the directrix, the parabola opens downwards. The distance from the vertex to the focus is 1. Thus, \(p = 1\).
The standard form of the equation is:
\[
(y - k) = -\frac{1}{4p}(x - h)^2
\]
Substituting \(h = -4\), \(k = 7\), and \(p = 1\):
\[
y - 7 = -\frac{1}{4}(x + 4)^2
\]
We can rearrange this to standard form \(y = -\frac{1}{4}(x + 4)^2 + 7\).
To match with \(y = -\frac{1}{4} x^2 - bx + 3\), we rewrite:
Taking the completed square form and equating terms, we need to fill in \(b\) where \(-b = 8\) from the standard form derived.
Thus, \(b = -2\).
**The answer is: b. -2**
---
### 3. Graph a parabola \(y = -\frac{1}{8} x^2 + \frac{3}{4} x + \frac{7}{8}\).
Let's find the missing x-value corresponding to the y-value of 0.
Set the equation to zero:
\[
0 = -\frac{1}{8} x^2 + \frac{3}{4} x + \frac{7}{8}
\]
To eliminate the fraction, multiply through by 8:
\[
0 = -x^2 + 6x + 7
\]
Rearranging gives:
\[
x^2 - 6x - 7 = 0
\]
Factoring:
\[
(x - 7)(x + 1) = 0
\]
Thus \(x = 7\) or \(x = -1\).
The missing x-value between -1 and 3 is \(7\).
**The answer is: a. 7**
---
### 4. The y-intercept of the parabola \(y = (x - \frac{3}{2})^2 + \frac{5}{4}\).
To find the y-intercept, set \(x = 0\):
\[
y = (0 - \frac{3}{2})^2 + \frac{5}{4} = \frac{9}{4} + \frac{5}{4} = \frac{14}{4} = \frac{7}{2} = 3.5
\]
**The answer is: c. 3.5**
---
### 5. The maximum height of the golf ball represented in the context.
The maximum height is represented by the **vertex**, which is the highest point of the quadratic function.
**The answer is: Option #1: vertex**
---
### 6. Identify the number of sides of the two-dimensional vertical cross-section of a pyramid.
The two-dimensional vertical cross-section of a pyramid has **3 sides** (forming a triangle).
**The answer is: b. 3**
---
### 7. Shape that can be rotated to create a three-dimensional object like an orange.
A **hemisphere** can be rotated to create a three-dimensional object like an orange.
**The answer is: c. 3**