Question 1

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$$y ^ { 2 } - 12 y - x + 32 = 0$$
A)Yes
B)No

Accepted Answer

The given equation is in the form of a parabola that opens left or right since the variable $y$ is squared. The domain of such a parabola is all real numbers if it opens up or down, but in this case, since it opens left or right, the domain is restricted. Therefore, the statement about determining the domain and range solely based on the vertex and the direction of opening is incorrect, as additional information or transformation into the standard form is needed to accurately determine these characteristics.

Question 2

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate
system and finding points of intersection.
-$$\begin{array} { l } 
x = ( y + 10 ) ^ { 2 } - 1 \
( x - 10 ) ^ { 2 } + ( y + 10 ) ^ { 2 } = 1
\end{array}$$

A) $\{ ( 10 , - 10 ) \}$
B) $\{ ( - 1 , - 10 ) \}$
C) $\{ ( - 1 , - 10 ) , ( 10 , - 10 ) \}$
D) $\varnothing$

Accepted Answer

The answer of Find the solution set for the system...

Question 3

Graph Parabolas with Vertices Not at the Origin
Find the vertex, focus, and directrix of the parabola with the given equation.
-$$( x - 1 ) ^ { 2 } = - 7 ( y + 2 )$$

A)

B)

C)

D)

Accepted Answer

The answer of Graph Parabolas with Vertices Not at the...

Question 4

Solve the problem.
-An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers
Are both 12.25 inches tall and stand 70 inches apart. Find the vertical distance from the roadway to the
Cable at a point on the road 17.5 inches from the lowest point of the cable.
A)3.06 in.
B)12.25 in.
C)3.26 in.
D)2.86 in.

Accepted Answer

The problem describes a scenario that can be modeled using the properties of a parabola. The cable forms a parabola with its vertex at the lowest point between the towers. Given the distance between the towers and their height, we can determine the parabola's equation or directly calculate the vertical distance at a specific point using principles of parabolic shapes. The correct calculation results in a vertical distance of 3.06 inches from the roadway to the cable at the specified point.

Question 5

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$$y = x ^ { 2 } + 8 x + 22$$
A)Yes
B)No

Accepted Answer

The parabola opens upward, so the vertex is the minimum point. The domain is all real numbers, and the range is [22, ∞).

Question 6

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$y ^ { 2 } - 12 y - x + 38 = 0$
A) Domain: $( 2 , \infty ]$
Range: $( - \infty , \infty )$
B) Domain: $( - \infty , - 2 )$
Range: $( - \infty , \infty )$
C) Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$
D) Domain: $( - \infty , \infty )$
Range: $( - \infty , - 2 ]$

Accepted Answer

The given equation can be rewritten as $(y-6)^2 = x-2$, which is the standard form of a parabolic equation with the vertex at $(2,6)$ and the direction opening rightward. The domain is therefore $(2,\infty]$ since the parabola only extends to the right. The range is $(-\infty,\infty)$ since the parabola extends infinitely up and down. Choice A is the only option that matches these characteristics.

Question 7

Solve the problem.
-A satellite dish is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 10 feet and a depth of 7 feet. The parabola is positioned in a rectangular coordinate system with its vertex at the origin. The receiver should be placed at the focus $( 0 , \mathrm { p } )$. The value of $\mathrm { p }$ is given by the equation $\mathrm { a } = \frac { 1 } { 4 \mathrm { p } }$. How far from the base of the dish should the receiver be placed?

A) $\frac { 25 } { 28 }$ feet from the base
B) $3 \frac { 4 } { 7 }$ feet from the base
C) $\frac { 7 } { 25 }$ feet from the base
D) $1 \frac { 3 } { 25 }$ feet from the base

Accepted Answer

The answer of Solve the problem.
-A satellite dish is in...

Question 8

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate
system and finding points of intersection.
-$$\begin{aligned}
( y - 4 ) ^ { 2 } & = x + 16 \
y & = - \frac { 1 } { 4 } x
\end{aligned}$$

A) $\{ ( - 16,4 ) , ( 0,0 ) \}$
B) $\{ ( 16 , - 4 ) , ( 0,0 ) \}$
C) $\{ ( - 16,0 ) , ( 4,0 ) \}$
D) $\{ ( - 16,4 ) \}$

Accepted Answer

The answer of Find the solution set for the system...

Question 9

Solve the problem.
-An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers
Stand 40 inches apart. At a point between the towers and 10 inches along the road from the base of one
Tower, the cable is 1 inches above the roadway. Find the height of the towers.
A)4 in.
B)4.5 in.
C)3.5 in.
D)6 in.

Accepted Answer

The cable forms a parabola. Using the vertex form of a parabola equation, the vertex is at the lowest point (10, 1). The symmetry of the parabola means the towers are equidistant from the vertex, so they are 20 inches from the vertex. Using the equation of the parabola, the height of the towers is 4 inches.

Question 10

Solve the problem.
-A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 170 feet and a maximum height of 30 feet. Find the height of the arch at 15 feet from its center.
A)29.1 ft
B)0.2 ft
C)3.7 ft
D)21.3 ft

Accepted Answer

The equation of a parabolic arch can be represented as $y = ax^2 + bx + c$. Given the maximum height of the arch is 30 feet, and it spans 170 feet, we can assume the vertex of the parabola is at the center of the span, making the coordinate of the vertex (0, 30) since the maximum height is at the center. The parabola crosses the x-axis at $x = \pm 85$ feet (half of the span on either side of the center), which are the roots of the equation, meaning $y = 0$ at these points.To find the equation of the parabola, we use the vertex form of a parabolic equation, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Here, $h = 0$ and $k = 30$, so the equation simplifies to $y = ax^2 + 30$. Since the parabola crosses the x-axis at $x = 85$ and $x = -85$, we can use one of these points to solve for $a$. Plugging $x = 85$ and $y = 0$ into the equation gives us $0 = a(85)^2 + 30$. Solving for $a$ gives us $a = -30/7225$.To find the height of the arch 15 feet from its center, we substitute $x = 15$ into the equation $y = a(x)^2 + 30$, using the $a$ we just found. This gives us $y = -30/7225 \cdot 15^2 + 30$, which calculates to approximately 29.1 feet. Therefore, the height of the arch at 15 feet from its center is approximately 29.1 feet.

Question 11

Solve the problem.
-An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers
Are both 4 inches tall and stand 40 inches apart. At some point along the road from the lowest point of the
Cable, the cable is 0.36 inches above the roadway. Find the distance between that point and the base of the
Nearest tower.
A)14 in.
B)5.8 in.
C)14.2 in.
D)6.2 in.

Accepted Answer

The distance from the lowest point of the cable to the roadway is half the distance between the towers, or 20 inches. The distance from the lowest point of the cable to the point where it is 0.36 inches above the roadway is 20 - 0.36 = 19.64 inches. The distance from the point where the cable is 0.36 inches above the roadway to the base of the nearest tower is 19.64 - 4 = 15.64 inches.

Question 12

Additional Concepts
Determine the direction in which the parabola opens, and the vertex.
-$$y = x ^ { 2 } - 6 x + 5$$
A) Opens upward; $( 3 , - 4 )$
B) Opens upward; $( - 3 , - 4 )$
C) Opens to the right; $( - 4,3 )$
D) Opens to the right; $( - 4 , - 3 )$

Accepted Answer

To determine the direction of the parabola, we look at the coefficient of the squared term. In this case, the coefficient is positive (1), so the parabola opens upward.

To find the vertex, we can use the formula:

$$ \left( \frac{-b}{2a}, f \left( \frac{-b}{2a} \right) \right) $$

where a is the coefficient of the squared term, b is the coefficient of the linear term, and f is the function itself.

Plugging in the values from the given equation, we get:

$$ \left( \frac{-(-6)}{2(1)}, f \left( \frac{-(-6)}{2(1)} \right) \right) $$
$$ \left( 3, f(3) \right) $$
$$ \left( 3, (3)^2 - 6(3) + 5 \right) $$
$$ \left( 3,-4 \right) $$

Therefore, the correct answer is A.

Question 13

Graph Parabolas with Vertices Not at the Origin
Find the vertex, focus, and directrix of the parabola with the given equation.
-$$( x - 2 ) ^ { 2 } = 7 ( y + 1 )$$

A)

B)

C)

D)

Accepted Answer

The answer of Graph Parabolas with Vertices Not at the...

Question 14

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$$x = - ( y - 8 ) ^ { 2 } - 1$$
A) Domain: $( - \infty , - 1 ]$
Range: $( - \infty , \infty )$ 
B) Domain: $( - \infty , - 1 )$
Range: $( - \infty , \infty )$
C) Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$
D) Domain: $( - \infty , \infty )$ 
Range: $( - \infty ,-1)$ Is the relation a function?

Accepted Answer

The vertex is given by the expression $-(y-8)^2-1$, which indicates that the vertex is at the point $( -1, 8 )$. Since the parabola opens downward (the coefficient of the squared term is negative), the highest point (the vertex) is the maximum value of the function, and the range is from negative infinity to the vertex, which is $( - \infty , 8 ]$. The domain is from negative infinity to the x-value of the vertex, which is $( - \infty , -1 ]$. Therefore, the correct choice is A.

As for whether the relation is a function, we note that since there is only one value of x for each value of y, the relation passes the vertical line test and is a function.

Question 15

Additional Concepts
Determine the direction in which the parabola opens, and the vertex.
-$$x = - ( y - 8 ) ^ { 2 } + 1$$
A) Opens to left; $( 1,8 )$
B) Opens to left; $( 1 , - 8 )$
C) Opens to left; $( 8,1 )$
D) Opens to right; $( 8,1 )$

Accepted Answer

To determine the direction in which the parabola opens, look at the coefficient of the squared term. In this case, it is negative, which means the parabola opens to the left. 
To find the vertex, we need to rewrite the equation in vertex form: 
$$x + (y-8)^2 = 1$$
$$(y-8)^2 = -x+1$$
$$y-8 = \pm \sqrt{-x+1}$$
$$y = \pm \sqrt{-x+1}+8$$
The vertex is at the point where the square root term is equal to 0, which is when $-x+1 = 0$ or $x=1$. So the vertex is at $(1,8)$.

Question 16

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$$x = - ( y + 1 ) ^ { 2 } + 8$$
A)Yes
B)No

Accepted Answer

The given equation is in the form $x = - ( y + 1 ) ^ { 2 } + 8$, which represents a parabola that opens to the left (since the coefficient of the squared term is negative). This means the parabola's domain is all real numbers (since it extends infinitely in the y-direction), but its range is restricted by the vertex. The vertex of this parabola is at $(-1, 8)$, so the range is all real numbers, but the domain is restricted to $x \leq 8$. The question seems to misunderstand the relation's domain and range based on the parabola's orientation, hence the correct choice is "No."

Question 17

Solve the problem.
-A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 25 feet. If the distance across the top of the mirror is 58 inches, how deep is the mirror in the center?
A) $\frac { 841 } { 1200 } \mathrm { in }$.
B) $\frac { 841 } { 100 } \mathrm { in }$.
C) $\frac { 841 } { 14400 }$ in.
D) $\frac { 625 } { 116 } \mathrm { in }$.

Accepted Answer

Let the depth of the mirror be $d$ inches. Using the mirror equation $\frac{1}{f}=\frac{1}{p}+\frac{1}{q}$, where $p$ is the distance from the vertex to the object and $q$ is the distance from the vertex to the image, we have $f=q=25$ feet $=300$ inches and $p=300-d$ inches. We can now solve for $d$: $$\frac{1}{300}=\frac{1}{300-d}+\frac{1}{300}$$ $$\frac{1}{300-d}=\frac{1}{600}$$ $$d=300-\frac{300}{2}=\frac{300}{2}=150	ext{ inches}$$ Therefore, the answer is $\frac{841}{1200}$ inches or about $0.701$ inches.

Question 18

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.
-$$y = x ^ { 2 } + 2 x - 4$$
A) Domain: $( - \infty , \infty )$
Range: $[ - 5 , \infty )$
B) Domain: $( - \infty , \infty )$
Range: $( - 5 , \infty )$
C) Domain: $( - 5 , \infty )$
Range: $( - \infty , \infty )$
D) Domain: $[ - 5 , \infty )$
Range: $( - \infty , \infty )$

Accepted Answer

The parabola opens upwards (since the coefficient of $x^2$ is positive), and its vertex can be found using the formula $-\frac{b}{2a}$ for the x-coordinate. Here, $a = 1$ and $b = 2$, so the vertex's x-coordinate is $-\frac{2}{2(1)} = -1$. Plugging this back into the equation gives the y-coordinate of the vertex as $(-1)^2 + 2(-1) - 4 = -5$. Therefore, the vertex is $(-1, -5)$, and since the parabola opens upwards, the domain is all real numbers $(-\infty, \infty)$, and the range is $[-5, \infty)$ because the vertex represents the lowest point on the parabola.

Question 19

Additional Concepts
Determine the direction in which the parabola opens, and the vertex.
-$$y ^ { 2 } - 4 y - x - 1 = 0$$
A) Opens to the right; $( - 5,2 )$
B) Opens to the left; $( 5 , - 2 )$
C) Opens downward; $( 2,5 )$
D) Opens downward; $( - 2,5 )$

Accepted Answer

To determine the direction and vertex, first rewrite the equation in a standard form. Rearrange the given equation to $x = y^2 - 4y - 1$. Completing the square on the $y$ terms gives $x = (y - 2)^2 - 5$, indicating the parabola opens to the right with vertex $(-5, 2)$.

Question 20

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate
system and finding points of intersection.
-$$\begin{array} { l } 
x = y ^ { 2 } - 2 \
x = y ^ { 2 } - 2 y
\end{array}$$

A) $\{ ( - 1,1 ) \}$
B) $\{ ( 1 , - 1 ) \}$
C) $\{ ( 1,1 ) \}$
D) $\{ ( - 1 , - 1 ) \}$

Accepted Answer

The answer of Find the solution set for the system...

Quiz 7: Conic Sections

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y ^ { 2 } - 12 y - x + 32 = 0$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y = x ^ { 2 } + 8 x + 22$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y ^ { 2 } - 12 y - x + 38 = 0$

Solve the problem. -A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 170 feet and a maximum height of 30 feet. Find the height of the arch at 15 feet from its center.

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $y = x ^ { 2 } - 6 x + 5$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $x = - ( y - 8 ) ^ { 2 } - 1$

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $x = - ( y - 8 ) ^ { 2 } + 1$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $x = - ( y + 1 ) ^ { 2 } + 8$

Solve the problem. -A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 25 feet. If the distance across the top of the mirror is 58 inches, how deep is the mirror in the center?

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y = x ^ { 2 } + 2 x - 4$

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $y ^ { 2 } - 4 y - x - 1 = 0$

Quiz 7: Conic Sections

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - y2−12y−x+32=0y ^ { 2 } - 12 y - x + 32 = 0y2−12y−x+32=0

Graph Parabolas with Vertices Not at the Origin Find the vertex, focus, and directrix of the parabola with the given equation. - (x−1) 2=−7(y+2) ( x - 1 ) ^ { 2 } = - 7 ( y + 2 ) (x−1) 2=−7(y+2)

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - y=x2+8x+22y = x ^ { 2 } + 8 x + 22y=x2+8x+22

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - y2−12y−x+38=0y ^ { 2 } - 12 y - x + 38 = 0y2−12y−x+38=0

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. - (y−4) 2=x+16y=−14x\begin{aligned}( y - 4 ) ^ { 2 } & = x + 16 \\y & = - \frac { 1 } { 4 } x\end{aligned}(y−4) 2y​=x+16=−41​x​

Solve the problem. -A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 170 feet and a maximum height of 30 feet. Find the height of the arch at 15 feet from its center.

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - y=x2−6x+5y = x ^ { 2 } - 6 x + 5y=x2−6x+5

Graph Parabolas with Vertices Not at the Origin Find the vertex, focus, and directrix of the parabola with the given equation. - (x−2) 2=7(y+1) ( x - 2 ) ^ { 2 } = 7 ( y + 1 ) (x−2) 2=7(y+1)

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - x=−(y−8) 2−1x = - ( y - 8 ) ^ { 2 } - 1x=−(y−8) 2−1

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - x=−(y−8) 2+1x = - ( y - 8 ) ^ { 2 } + 1x=−(y−8) 2+1

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - x=−(y+1) 2+8x = - ( y + 1 ) ^ { 2 } + 8x=−(y+1) 2+8

Solve the problem. -A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 25 feet. If the distance across the top of the mirror is 58 inches, how deep is the mirror in the center?

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - y=x2+2x−4y = x ^ { 2 } + 2 x - 4y=x2+2x−4

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - y2−4y−x−1=0y ^ { 2 } - 4 y - x - 1 = 0y2−4y−x−1=0

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. - x=y2−2x=y2−2y\begin{array} { l } x = y ^ { 2 } - 2 \\x = y ^ { 2 } - 2 y\end{array}x=y2−2x=y2−2y​

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y ^ { 2 } - 12 y - x + 32 = 0$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y = x ^ { 2 } + 8 x + 22$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y ^ { 2 } - 12 y - x + 38 = 0$

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $y = x ^ { 2 } - 6 x + 5$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $x = - ( y - 8 ) ^ { 2 } - 1$

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $x = - ( y - 8 ) ^ { 2 } + 1$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $x = - ( y + 1 ) ^ { 2 } + 8$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. - $y = x ^ { 2 } + 2 x - 4$

Additional Concepts Determine the direction in which the parabola opens, and the vertex. - $y ^ { 2 } - 4 y - x - 1 = 0$