CTJan27 Online Math Methods - Hyperbolas

Choose the best answer for each question.
ℹ️

The basic equation y = k/x and its characteristic shape

Questions for: The basic equation y = k/x and its characteristic shape

1

What is the general shape of the graph of the equation $y = \frac{k}{x}$, where $k$ is a non-zero constant?

A straight line

An ellipse

A rectangular hyperbola

A circle

A parabola
2

If the equation of a hyperbola is $y = \frac{5}{x}$, in which quadrants does its graph lie?

Quadrants I and III

Quadrants I and IV

Quadrants II and IV

Quadrants II and III

Quadrants I and II
3

The graph of the equation $y = \frac{k}{x}$ lies in Quadrants II and IV. What can be concluded about the value of the constant $k$?

$k = 1$

$k < 0$

$k = 0$

$k$ can be any real number

$k > 0$
4

What are the equations of the asymptotes for the graph of $y = \frac{-2}{x}$?

The graph has no asymptotes

$x = -2$ and $y = 0$

$x = 0$ and $y = -2$

$x = 2$ and $y = -2$

$x = 0$ and $y = 0$
5

Which of the following points lies on the graph of the hyperbola $y = \frac{12}{x}$?

$(4, 8)$

$(6, 3)$

$(2, 8)$

$(3, 4)$

$(1, 11)$
ℹ️

Identifying vertical asymptotes from the equation

Questions for: Identifying vertical asymptotes from the equation

6

What is the equation of the vertical asymptote for the rectangular hyperbola given by the equation $y = \frac{5}{x-3} + 2$?

$y = 3$

$x = 5$

$y = 2$

$x = 3$

$x = -3$
7

Identify the vertical asymptote for the function $f(x) = \frac{-2}{x+7}$.

$x = -2$

$y = -7$

$x = -7$

$y = 0$

$x = 7$
8

Find the equation of the vertical asymptote for the graph of $y = \frac{4}{2x-8} - 1$.

$x = 4$

$x = -4$

$x = 2$

$x = 8$

$y = -1$
9

What is the vertical asymptote of the hyperbola defined by the equation $y = \frac{1}{6-x}$?

$y = 0$

$x = -6$

$y = 6$

$x = 1$

$x = 6$
10

For a rectangular hyperbola with the general equation $y = \frac{a}{x-h} + k$, where $a eq 0$, what is the equation of the vertical asymptote?

$x = -h$

$x = h$

$x = a$

$y = h$

$y = k$
ℹ️

Identifying horizontal asymptotes from the equation

Questions for: Identifying horizontal asymptotes from the equation

11

What is the equation of the horizontal asymptote for the rectangular hyperbola with the equation $y = \frac{2}{x-3} + 5$?

$y = 3$

$x = 5$

$y = 5$

$y = 2$

$x = 3$
12

Identify the horizontal asymptote of the function $f(x) = \frac{-4}{x+1} - 7$.

$x = -1$

$y = -1$

$y = 7$

$y = -7$

$x = -7$
13

Determine the equation of the horizontal asymptote for the graph of $y = \frac{5}{x-6}$.

$y = 0$

$x = 6$

No horizontal asymptote

$y = 5$

$y = 6$
14

What is the horizontal asymptote of the hyperbola defined by the equation $y = 9 + \frac{1}{x+2}$?

$y = 1$

$x = -2$

$x = 9$

$y = -2$

$y = 9$
15

Find the equation of the horizontal asymptote for the function $y = -11 + \frac{8}{x}$.

$x = 0$

$y = -11$

$y = 0$

$y = 8$

$x = -11$
ℹ️

The effect of horizontal and vertical translations (h and k)

Questions for: The effect of horizontal and vertical translations (h and k)

16

What are the equations of the vertical and horizontal asymptotes for the rectangular hyperbola with the equation $y = \frac{3}{x-2} + 5$?

Vertical asymptote: $x = 2$, Horizontal asymptote: $y = -5$

Vertical asymptote: $x = -5$, Horizontal asymptote: $y = -2$

Vertical asymptote: $x = -2$, Horizontal asymptote: $y = -5$

Vertical asymptote: $x = 2$, Horizontal asymptote: $y = 5$

Vertical asymptote: $x = 5$, Horizontal asymptote: $y = 2$
17

A rectangular hyperbola has a vertical asymptote at $x = -4$ and a horizontal asymptote at $y = 1$. Which of the following could be the equation of this hyperbola?

$y = \frac{2}{x+4} + 1$

$y = \frac{2}{x-4} + 1$

$y = \frac{2}{x+1} + 4$

$y = \frac{2}{x-1} - 4$

$y = \frac{2}{x+4} - 1$
18

How is the graph of $y = \frac{1}{x+3} - 7$ translated from the parent graph of $y = \frac{1}{x}$?

7 units to the left and 3 units down

7 units to the right and 3 units up

3 units to the left and 7 units down

3 units to the left and 7 units up

3 units to the right and 7 units down
19

What are the coordinates of the center (the point of intersection of the asymptotes) of the hyperbola given by the equation $y = \frac{-5}{x-1} + 6$?

$(1, -6)$

$(6, 1)$

$(-1, -6)$

$(-1, 6)$

$(1, 6)$
20

The graph of the function $f(x) = \frac{2}{x}$ is translated to create the graph of $g(x) = \frac{2}{x-h} + k$. The new graph has a vertical asymptote at $x=5$ and a horizontal asymptote at $y=-3$. What is the equation for $g(x)$?

$g(x) = \frac{2}{x+3} - 5$

$g(x) = \frac{2}{x-5} - 3$

$g(x) = \frac{2}{x-5} + 3$

$g(x) = \frac{2}{x+5} - 3$

$g(x) = \frac{2}{x-3} + 5$
ℹ️

Determining graph quadrants based on the sign of the numerator

Questions for: Determining graph quadrants based on the sign of the numerator

21

The graph of the function $y = \frac{5}{x}$ lies in which quadrants of the Cartesian plane?

Quadrants 2 and 3

Quadrants 1 and 2

Quadrants 1 and 3

Quadrants 3 and 4

Quadrants 2 and 4
22

The branches of the hyperbola with the equation $f(x) = \frac{-10}{x}$ are located in which quadrants?

Quadrants 1 and 2

Quadrants 1 and 3

Quadrants 1 and 4

Quadrants 3 and 4

Quadrants 2 and 4
23

Consider the function $g(x) = \frac{3}{x-2} + 1$. Relative to its asymptotes ($x=2$ and $y=1$), in which 'quadrants' do the branches of the graph lie?

Quadrants 2 and 4

Quadrants 1 and 3

Quadrants 2 and 3

Quadrants 1 and 2

Quadrants 3 and 4
24

The graph of the equation $y = -\frac{7}{x+4} - 5$ has its branches in which quadrants, relative to its asymptotes?

Quadrants 1 and 3

Quadrants 3 and 4

Quadrants 2 and 3

Quadrants 2 and 4

Quadrants 1 and 2
25

A rectangular hyperbola is described by the equation $y = \frac{k}{x}$. If the graph passes through the point $(-2, 3)$, in which quadrants do the branches of the hyperbola lie?

Quadrants 2 and 4, because $k$ must be negative.

It cannot be determined from the given information.

Quadrants 1 and 3, because $k$ must be positive.

Quadrants 3 and 4, because the y-coordinate is positive.

Quadrants 1 and 2, because the x-coordinate is negative.
ℹ️

Calculating x-intercepts

Questions for: Calculating x-intercepts

26

Find the x-intercept of the rectangular hyperbola with the equation $y = \frac{2}{x-3} + 1$.

$x=5$

$x=1$

$x=3$

$x=\frac{1}{3}$

No x-intercept
27

What is the x-intercept of the function $f(x) = \frac{4}{x+2} - 2$?

$x=-4$

$x=1$

$x=4$

$x=-2$

$x=0$
28

Determine the coordinates of the x-intercept for the graph of $y = -\frac{6}{x-1} - 3$.

$(1, 0)$

$(3, 0)$

$(-3, 0)$

$(-1, 0)$

$(0, 3)$
29

Calculate the x-intercept for the hyperbola given by the equation $y = \frac{5}{2x+4} + 1$.

$x = -\frac{9}{2}$

$x = \frac{1}{2}$

$x = -2$

$x = \frac{9}{2}$

$x = \frac{9}{4}$
30

Which of the following rectangular hyperbolas does not have an x-intercept?

$y = \frac{3}{x-2} + 1$

$y = \frac{1}{x+6} - 6$

$y = \frac{5}{x+1}$

$y = 3 + \frac{1}{x-3}$

$y = \frac{-2}{x} - 4$
ℹ️

Calculating y-intercepts

Questions for: Calculating y-intercepts

31

Find the y-intercept of the rectangular hyperbola with the equation $y = \frac{4}{x-2} + 5$.

$(2, 5)$

$(0, 7)$

$(0, 5)$

$(0, 3)$

$(0, -2)$
32

What are the coordinates of the y-intercept for the function $f(x) = \frac{-6}{x+3} - 1$?

$(0, -1)$

$(0, -3)$

$(0, -2)$

$(-3, -1)$

$(0, 1)$
33

Determine the y-intercept of the graph of the equation $y = \frac{4}{x+2} - 2$.

$(-2, -2)$

$(0, 2)$

$(0, -2)$

No y-intercept

$(0, 0)$
34

A rectangular hyperbola is defined by the equation $y = \frac{3x+1}{x-1}$. At which point does it cross the y-axis?

$(0, -1)$

$(0, 3)$

$(1, 0)$

$(0, 1)$

$(0, -3)$
35

A rectangular hyperbola with the equation $y = \frac{a}{x-2} + k$ has a y-intercept at $(0, 1)$ and passes through the point $(4, 5)$. What is the value of $k$?

$k=2$

$k=5$

$k=3$

$k=4$

$k=1$
ℹ️

Stating the domain and range

Questions for: Stating the domain and range

36

Find the domain of the function $f(x) = \frac{2}{x-3} + 5$.

$x \in \mathbb{R}, x eq 3$

$x \in \mathbb{R}, x eq 5$

$y \in \mathbb{R}, y eq 5$

$x \in \mathbb{R}, x eq -3$

$x > 3$
37

Find the range of the function $g(x) = \frac{-1}{x+4} - 2$.

$y \in \mathbb{R}, y eq 2$

$y < -2$

$x \in \mathbb{R}, x eq -4$

$y \in \mathbb{R}, y eq -2$

$y \in \mathbb{R}, y eq -4$
38

What is the domain of the rectangular hyperbola given by the equation $y = \frac{5}{x} + 1$?

$x > 0$

$x \in \mathbb{R}$

$x \in \mathbb{R}, x eq 1$

$y \in \mathbb{R}, y eq 1$

$x \in \mathbb{R}, x eq 0$
39

What is the range of the rectangular hyperbola given by the equation $y = \frac{7}{x - 6}$?

$y \in \mathbb{R}$

$y \in \mathbb{R}, y eq 0$

$x \in \mathbb{R}, x eq 6$

$y \in \mathbb{R}, y eq 6$

$y \in \mathbb{R}, y eq -6$
40

State the domain and range of the function $h(x) = \frac{1}{x+1} + 1$.

Domain: $x \in \mathbb{R}$; Range: $y \in \mathbb{R}, y eq 1$

Domain: $x \in \mathbb{R}, x eq -1$; Range: $y \in \mathbb{R}, y eq 1$

Domain: $x \in \mathbb{R}, x eq -1$; Range: $y \in \mathbb{R}$

Domain: $y \in \mathbb{R}, y eq 1$; Range: $x \in \mathbb{R}, x eq -1$

Domain: $x \in \mathbb{R}, x eq 1$; Range: $y \in \mathbb{R}, y eq -1$
41

A rectangular hyperbola has the equation $y = 8 - \frac{3}{x-2}$. What is its domain?

$x \in \mathbb{R}$

$x \in \mathbb{R}, x eq 8$

$x \in \mathbb{R}, x eq 2$

$x \in \mathbb{R}, x eq -2$

$y \in \mathbb{R}, y eq 8$
42

A rectangular hyperbola has the equation $y = \frac{10}{5-x} + 7$. What is its range?

$y \in \mathbb{R}, y eq 5$

$x \in \mathbb{R}, x eq -5$

$x \in \mathbb{R}, x eq 5$

$y \in \mathbb{R}, y eq 7$

$y \in \mathbb{R}, y eq -7$
43

The function $f(x) = \frac{k}{x-p} + q$ has a domain of $x \in \mathbb{R}, x eq -5$ and a range of $y \in \mathbb{R}, y eq 10$. What are the values of $p$ and $q$?

$p = 10, q = -5$

$p = -5, q = -10$

$p = -5, q = 10$

$p = 5, q = -10$

$p = 5, q = 10$
44

Find the domain and range for the function $y = \frac{2x+1}{x-1}$.

Domain: $x \in \mathbb{R}, x eq 1$; Range: $y \in \mathbb{R}, y eq 2$

Domain: $y \in \mathbb{R}, y eq 2$; Range: $x \in \mathbb{R}, x eq 1$

Domain: $x \in \mathbb{R}, x eq -1$; Range: $y \in \mathbb{R}, y eq -2$

Domain: $x \in \mathbb{R}, x eq 1$; Range: $y \in \mathbb{R}, y eq 1$

Domain: $x \in \mathbb{R}, x eq -1$; Range: $y \in \mathbb{R}, y eq 2$
45

Which of the following functions has a domain of $x \in \mathbb{R}, x eq 4$ and a range of $y \in \mathbb{R}, y eq -3$?

$y = \frac{1}{x+4} - 3$

$y = \frac{1}{x-4} + 3$

$y = \frac{1}{x+3} - 4$

$y = \frac{1}{x-3} + 4$

$y = \frac{1}{x-4} - 3$
ℹ️

Determining the equation from a graph's key features

Questions for: Determining the equation from a graph's key features

46

A rectangular hyperbola has a vertical asymptote at $x=2$ and a horizontal asymptote at $y=-1$. The graph passes through the point $(3,1)$. Determine the equation of the hyperbola.

$y = \frac{2}{x-2} - 1$

$y = \frac{2}{x+2} + 1$

$y = \frac{-2}{x-2} - 1$

$y = \frac{1}{x-2} - 1$

$y = \frac{2}{x-3} - 1$
47

A rectangular hyperbola has a vertical asymptote at $x=-3$ and a horizontal asymptote at $y=4$. It passes through the origin $(0,0)$. What is its equation?

$y = \frac{12}{x+3} + 4$

$y = \frac{-3}{x+3} + 4$

$y = \frac{4}{x+3} + 4$

$y = \frac{-12}{x+3} + 4$

$y = \frac{-12}{x-3} + 4$
48

The graph of a hyperbola has asymptotes with equations $x=1$ and $y=-2$. The y-intercept of the graph is at $(0, -3)$. Determine the equation of the hyperbola.

$y = \frac{2}{x-1} - 2$

$y = \frac{1}{x+1} - 2$

$y = \frac{1}{x-1} - 2$

$y = \frac{-1}{x-1} - 2$

$y = \frac{1}{x-1} + 2$
49

A rectangular hyperbola has a horizontal asymptote at $y=5$ and an x-intercept at $(2, 0)$. The vertical asymptote is $x=3$. Find the equation of the hyperbola.

$y = \frac{-5}{x-3} + 5$

$y = \frac{5}{x-5} + 3$

$y = \frac{5}{x+3} + 5$

$y = \frac{3}{x-2} + 5$

$y = \frac{5}{x-3} + 5$
50

The asymptotes of a hyperbola are given by the lines $x=-4$ and $y=-1$. The graph passes through the point $(-5, -3)$. What is the equation of the hyperbola?

$y = \frac{4}{x+4} - 1$

$y = \frac{-2}{x+4} - 1$

$y = \frac{2}{x+4} - 1$

$y = \frac{2}{x-4} - 1$

$y = \frac{2}{x+4} + 1$
51

A hyperbola has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=3$. It passes through the point $(1, 5)$. Find its equation.

$y = \frac{1}{x} + 3$

$y = \frac{3}{x} + 2$

$y = \frac{2}{x+1} + 3$

$y = \frac{2}{x} + 3$

$y = \frac{2}{x} - 3$
52

The graph of a hyperbola has a y-intercept at $(0, 2.5)$ and an x-intercept at $(-5, 0)$. The vertical asymptote is the line $x=-2$. What is the equation of the hyperbola?

$y = \frac{3}{x-2} + 1$

$y = \frac{3}{x+2} + 1$

$y = \frac{6}{x+2} - 1$

$y = \frac{3}{x+2} - 2$

$y = \frac{-3}{x+2} + 1$
53

A hyperbola has asymptotes $y=1$ and $x=-1$. It passes through the point $(1, 2)$. What is the value of '$a$' in the standard equation $y = \frac{a}{x-p} + q$?

$a=-2$

$a=2$

$a=1$

$a=0.5$

$a=-1$
54

The horizontal asymptote of a hyperbola is $y=0$ and the vertical asymptote is $x=4$. It passes through the point $(2, -1)$. What is its equation?

$y = \frac{2}{x-4}$

$y = \frac{4}{x-2}$

$y = \frac{-2}{x-4}$

$y = \frac{1}{x-4}$

$y = \frac{2}{x+4}$
55

A rectangular hyperbola is defined by the equation $y=\frac{a}{x-p}+q$. Its asymptotes are $x=5$ and $y=-3$. The graph's x-intercept is $(4,0)$. Find the equation.

$y = \frac{5}{x-4} - 3$

$y = \frac{-3}{x-5} + 3$

$y = \frac{3}{x-5} - 3$

$y = \frac{-3}{x+5} - 3$

$y = \frac{-3}{x-5} - 3$

Password Protected