Deck 1: Analytic Geometry

Question 1

Find the domain of $f(x)=\frac{14}{\sqrt{4 x+30}}$ .

A) All real numbers $x<-7.5$
B) All real numbers $x>-7.5$
C) All real numbers $x \geq-7.5$
D) All real numbers $x \leq-7.5$

Answer 1

All real numbers $x>-7.5$

Question 2

Given the function $f(x)=3 x^{2}+7 x-4$ , find $f\left(-\frac{1}{2}\right)$ .

A) -1.25
B) 0.25
C) -6.75
D) -8.25

Answer 2

-6.75

Question 3

An object dropped from an airplane $3250 \mathrm{~m}$ above the ground falls according to the equation $\mathrm{h}=$ $3250-4.95 t^{2}$ , where $h$ is the height in metres above the ground and $t$ is the time in seconds. Find the time for the object to fall a height of $2755 \mathrm{~m}$ above the ground.

A) $8.0 \mathrm{~s}$
B) $1 \overline{0} \mathrm{~s}$
C) $17 \mathrm{~s}$
D) $2 \overline{0} \mathrm{~s}$

Answer 3

$1 \overline{0} \mathrm{~s}$

Question 4

Find the equation of the line that passes through $(-5,13)$ with slope -0.75.

A) $3 x+4 y=57$
B) $4 x+3 y=19$
C) $4 x+3 y=-59$
D) $3 x+4 y=67$

Question 5

Find the equation of a line that passes through $(8,-2)$ and is perpendicular to $3 x-5 y=16$ .

A) $5 x+3 y=34$
B) $3 x-5 y=34$
C) $3 x+5 y=34$
D) $5 x-3 y=34$

Question 6

Find the distance between the points (5, - 7) and (- 6, - 3). Round to three significant digits.

A) 10.2
B) 11.7
C) 10.0
D) 14.9

Question 7

Find the equation of a line that passes through $(-12,6)$ and is perpendicular to $4 x-3 y=8$ .

A) $3 x-4 y=60$
B) $3 x+4 y=12$
C) $-4 x+37=66$
D) $3 x+47=-12$

Question 8

Find the coordinates of the point midway between (4, - 6) and (-9, 14).

A) $(13,-20)$
B) $(-5,8)$
C) $(-2.5,4)$
D) $(6.5,-10)$

Question 9

Determine whether or not each relation is a function. Write its domain and range.

- $\{(-3,4),(-3,6),(-3,8),(-3,10)\}$

Question 10

Determine whether or not each relation is a function. Write its domain and range.

- $y=4 x^{2}-1$

Question 11

Determine whether or not each relation is a function. Write its domain and range.

- $y=5-\sqrt{6 x-12}$

Question 12

Determine whether or not each relation is a function. Write its domain and range.

- $x=7-y^{2}$

Question 13

Determine whether or not each relation is a function. Write its domain and range.

-Given $f(x)=\frac{3 x-x^{2}}{x}$ , find
a) $f(2)$ b) $\mathrm{f}(-4)$ c) $\mathrm{f}(3 \mathrm{a})$

Question 14

Graph each equation.

- $3 x-2 y=12$

Question 15

Graph each equation.

- $y=x^{3}-5 x^{2}+12$

Question 16

Find the slope of the line passing through the points $(-7,-4)$ and $(-1,-9)$ .

Question 17

Find the slope of the line perpendicular to $y=\frac{3}{5} x+2$ .

Question 18

Find the equation of the line that passes through (3, - 7) with slope- $\frac{1}{2}$ .

Question 19

Find the equation of a line that is perpendicular to the $x$ - axis and passes through the point $(5,-3)$ .

Question 20

Find the equation of a line that passes through the point $(3,-8)$ and is parallel to $2 x-y=6$ .

Question 21

Determine whether the given pair of equations represents lines which are parallel, perpendicular, or neither. $2 x-3 y-3=0$ and $3 x+2 y-8=0$

Question 22

Find the distance between the points (7, - 3) and (-5, - 8).

Question 23

Find the coordinates of the point midway between $(-3,-5)$ and $(7,-11)$ .

Question 24

Determine whether or not triangle $\mathrm{ABC}$ is a right triangle. A $(-1.6,4.2), B(8,9), C(5,-9)$

Question 25

Find the equation of the circle with center at $(-7,-2)$ and passing through the point $(-3,6)$ .

A) $x^{2}+y^{2}-6 x+12 y-35=0$
B) $x^{2}+y^{2}+6 x-12 y-35=0$
C) $x^{2}+y^{2}+14 x+4 y-27=0$
D) $x^{2}+y^{2}-14 x-4 y-27=0$

Question 26

Find the equation of the parabola with focus ( $-2,7$ ) and directrix $x=4$ .

A) $y^{2}-14 y+12 x+37=0$
B) $x^{2}+4 x-6 y+37=0$
C) $x^{2}-4 x+22 y+37=0$
D) $y^{2}+14 y+4 x+37=0$

Question 27

Find the length of the major axis of the ellipse $4 x^{2}+9 y^{2}=324$ .

A) 2
B) 18
C) $\sqrt{117}$
D) 9

Question 28

Find the equation of the ellipse with vertices at $(-20,0)$ and $(20,0)$ and foci at $(15,0)$ and $(-15,0)$ .

A) $\frac{x^{2}}{400}+\frac{y^{2}}{175}=1$
B) $\frac{x^{2}}{175}+\frac{y^{2}}{400}=1$
C) $\frac{x^{2}}{400}+\frac{y^{2}}{225}=1$
D) $\frac{x^{2}}{225}+\frac{y^{2}}{400}=1$

Question 29

Find the foci of the hyperbola $\frac{x^{2}}{121}-\frac{y^{2}}{81}=1$ .

A) $( \pm \sqrt{202}, 0)$
B) $( \pm 11,0)$
C) $( \pm 2 \sqrt{10}, 0)$
D) $(0 \pm \sqrt{202})$

Question 30

Write the equation of the ellipse with center at (3, 4), vertices at (10, 4) and (-4, 4), and foci at (7, 4) and $(-1,4)$ .

A) $\frac{(\mathrm{x}-3)^{2}}{49}+\frac{(\mathrm{y}-4)^{2}}{33}=1$
B) $\frac{(\mathrm{x}-3)^{2}}{49}+\frac{(\mathrm{y}-4)^{2}}{16}=1$
C) $\frac{(\mathrm{x}-3)^{2}}{33}+\frac{(\mathrm{y}-4)^{2}}{49}=1$
D) $\frac{(x+3)^{2}}{49}+\frac{(y+4)^{2}}{33}=1$

Question 31

Find the $x$ - coordinates(s) of the solution(s) to the system of equations: $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}-21=0$ and $\mathrm{y}=2 \mathrm{x}-3$ .

A) 3.83 and -0.627
B) 3.83
C) -0.627 and -4.65
D) -4.25 and -4.65

Question 32

Change the point $\left(5, \frac{2 \pi}{3}\right)$ in the polar coordinate system to the equivalent point in the rectangular coordinate system.

A) $(-2.5,-4.33)$
B) $(-2.5,4.33)$
C) $(4.33,-2.5)$
D) $(-4.33,2.5)$

Question 33

Find the equation of the circle (in standard form) with center at $(7,-8)$ and with radius 10.

A) $(x-7)^{2}+(y+8)^{2}=100$
B) $\sqrt{(x-7)^{2}+(y+8)^{2}}=100$
C) $(x+7)^{2}-(y-8)^{2}=100$
D) $(x+7)^{2}+(y-8)^{2}=100$

Question 34

Find the center of the circle $x^{2}+y^{2}+16 x+20 y+155=0$ .

A) $(-8,10)$
B) $(-8,-10)$
C) $(8,10)$
D) $(8,-10)$

Question 35

Find the equation of the parabola with focus at ( $0,-8)$ and with directrix $y=8$ .

A) $y^{2}=32 x$
B) $y^{2}=-32 x$
C) $x^{2}=-32 y$
D) $x^{2}=32 y$

Question 36

The height of a projectile fired vertically upward is given by $y=f(t)=680 t-16 t 2$ (initial velocity is $680 \mathrm{ft} / \mathrm{s})$ . Find the maximum height in $\mathrm{ft}$ .

A) 3631
B) 7225
C) 7054
D) 3527

Question 37

Find the equation of the ellipse with vertices of $(0,-18)$ and $(0,18)$ and minor axis of length 30.

A) $\frac{x^{2}}{225}+\frac{y^{2}}{99}=1$
B) $\frac{x^{2}}{225}+\frac{y^{2}}{324}=1$
C) $\frac{x^{2}}{99}+\frac{y^{2}}{225}=1$
D) $\frac{x^{2}}{324}+\frac{y^{2}}{225}=1$

Question 38

Write the equation of the hyperbola with foci at $(0,15)$ and $(0,-15)$ and the transverse axis of length 20.

A) $4 y^{2}-5 x^{2}=500$
B) $5 y^{2}-4 x^{2}=500$
C) $13 y^{2}-4 x^{2}=1300$
D) $4 y^{2}-13 x^{2}=1300$

Question 39

Find the equation of the parabola with vertex at (-5, - 4), focus at $(-5,1)$ and directrix $y=-9$ .

A) $(x-5)^{2}=40(y-4)$
B) $(x+5)^{2}=20(y+4)$
C) $(y+4)^{2}=20(x+5)$
D) $(x+5)^{2}=40(y+4)$

Question 40

Name the type of graph represented by $x^{2}-4 y 2-10 x-8 y+5=0$ .

A) ellipse
B) parabola
C) circle
D) hyperbola

Question 41

Find the $x$ - coordinate(s) of the solution of the system of equations $9 y^{2}-16 x^{2}=144$ and $y=-0.5 x+$ 4.

A) -1.2 and 3.82
B) 3.4 and 5.91
C) -3.82 and 1.2
D) 1.2

Question 42

Change $x^{2}+y^{2}-4 x-12 y=0$ into an equation in polar form.

A) $r=4 \cos \theta-12 \sin \theta$
B) $r=4 \cos \theta+12 \sin \theta$
C) $r^{2}=4 \cos \theta+12 \sin \theta$
D) $r=48 \cos \theta \sin \theta$

Question 43

Name the number of petals in the rose curve $r=5 \sin 4 \theta$ .

A) 10
B) 8
C) 4
D) 20

Question 44

Graph the circle with center at ( $-2,-3)$ and $r=4$ .

Question 45

Find the equation of the circle (in standard form) with center at ( $4,-1$ ) and radius 3.

Question 46

Find the equation of the circle (in standard form) with center at $(7,3)$ and passing through $(6,-2)$ .

Question 47

Find the center and radius of the given circle. $x^{2}+y^{2}-6 x+4 y-3=0$ .

Question 48

Find the focus and the directrix of the parabola $y^{2}=-20 x$ .

Question 49

Find the equation of the parabola with focus $\mathrm{F}(0,-3)$ and directrix $\mathrm{y}=3$ .

Question 50

Find the equation of the parabola with vertex at $(0,0)$ and directrix $y=-6$ .

Question 51

Find the vertex and equation of the axis of the parabola $y=2 x^{2}-7 x-15$ .

Question 52

For the problems below, find the vertices, foci, and lengths of the major and minor axis of each ellipse.

- $\frac{x^{2}}{9}+\frac{y^{2}}{64}=1$

Question 53

For the problems below, find the vertices, foci, and lengths of the major and minor axis of each ellipse.

- $25 x^{2}+144 y^{2}=3600$

Question 54

For the problems below, find the vertices, foci, and lengths of the major and minor axis of each ellipse.
-Find the equation of the ellipse with vertices at $( \pm 7,0)$ and foci at $( \pm 2 \sqrt{10}, 0)$ .

Question 55

For the problems below, find the vertices, foci, and lengths of the major and minor axis of each ellipse.
-Find the equation of the ellipse with vertices at $(0, \pm 8)$ , and with a minor axis of length 10 .

Question 56

For the problems below, find the vertices, foci, and lengths of the transverse and conjugate axis for each hyperbola. Find the equations of the asymptotes.

- $\frac{x^{2}}{36}-\frac{y^{2}}{64}=1$

Question 57

For the problems below, find the vertices, foci, and lengths of the transverse and conjugate axis for each hyperbola. Find the equations of the asymptotes.

- $\frac{y^{2}}{100}-\frac{x^{2}}{81}=1$

Question 58

For the problems below, find the vertices, foci, and lengths of the transverse and conjugate axis for each hyperbola. Find the equations of the asymptotes.
-Find the equation of the hyperbola with vertices $( \pm 9,0)$ and foci at $( \pm 12,0)$ .

Question 59

For the problems below, find the vertices, foci, and lengths of the transverse and conjugate axis for each hyperbola. Find the equations of the asymptotes.
-Find the equation of the hyperbola with foci at $(0, \pm 7)$ and transverse axis of length 10.

Question 60

For the problems below, find the equation of each curve from the given information.

-Parabola with vertex at $(3,4)$ , focus at $(3,6)$ and directrix $y=2$ .

Question 61

Ellipse with center at (-3, 7), vertices at (-3, 20) and (-3, - 6) and minor axis of length 10.

Question 62

Name the graph of the equation $4 x^{2}-9 y^{2}+8 x+36 y-68=0$ . Locate the vertex if it is a parabola or the center if it is an ellipse or hyperbola.

Question 63

Determine whether the following equation represents a circle, a parabola, an ellipse, or a hyperbola. $\mathrm{x}^{2}+\mathrm{y}^{2}+10 \mathrm{x}-6 \mathrm{y}-15=0$ .

Question 64

Use the substitution method to find the real solutions for the following system of equations. Round to two significant digits. $\frac{x^{2}}{16}-\frac{y^{2}}{25}=1$ and $(x-3)^{2}+y^{2}=9$

Question 65

For the point $\left(4,45^{\circ}\right)$ , name three other sets of polar coordinates such that - $360^{\circ} \leq \theta \leq 360^{\circ}$ .

Question 66

Graph by plotting points. $\mathrm{r}=6-6 \sin \theta$

Question 67

Change $\left(5,60^{\circ}\right)$ to rectangular coordinates.

Question 68

Change(- $4 \sqrt{2},-4 \sqrt{2})$ to polar coordinates in radians.

Question 69

Change $x^{2}+y^{2}=49$ to polar form.

Question 70

Changer $=\frac{-5}{1+\sin \theta}$ to rectangular form.

Question 71

$r=7 \sin \theta$

Question 72

$r=8+4 \cos \theta$

Question 73

$r=5+12 \sin \theta$

Question 74

$r=3 \sin 5 \theta$

Question 75

$r=6 \cos 4 \theta$