Pre-Calculus Midterm Exam Review

I’m excited!

Is the graph a function or a relation?

Function Function

Relation

State the domain of the function:

€

y =x

1− x 2

€

y =x −1

x 2 − 9

€

y =x

x − 5

All real numbers except 1 or -1

All real numbers except 3 or -3

All real numbers except 5

€

y =x

x 2 − 5x All real numbers except 0 and 5

Find the composition functions below:

€

f (x) = 2x − 5

g(x) = x 2

€

( f og)(x) =

(g o f )(x) =

€

f (x) = 2x 2 + x − 2

g(x) = x − 3

€

( f og)(x) =

(g o f )(x) =

€

2x 2 − 5

€

(2x − 5)2

(2x − 5)(2x − 5)

4x 2 − 20x + 25€

2(x − 3)2 + (x − 3) − 2

2(x 2 − 6x + 9) + x − 5

2x 2 −12x +18 + x − 5

2x 2 −11x +13

€

(2x 2 + x − 2) − 3

2x 2 + x − 5

Find the x- and y- intercepts:

€

x + 2y −12 = 0

€

−4x + 6y + 24 = 0

(12,0) and (0,6) (6,0) and (0,-4)

Find the zero of each function:

€

f (x) = 3x − 2

€

f (x) = −12x 2 − 48

€

2

3

€

0 = −12x 2 − 48

48 = −12x 2

−4 = x 2

x = −4

x = ±2i

Dominic is opening a bank. He determined that he will need $22,000 to buy a building and supplies to start. He expects expenses for each following

month to be $12,300. Write an equation that models the total

expense y after x months.

€

y =12,300x + 22,000

Determine whether the graphs of the pair of equations are parallel,

coinciding, or neither.

x - 2y = 12 and 4x + y = 20 3x - 2y = -6 and 6x - 4y = -12

€

y =1

2x − 6

€

y = −4x + 20

€

y =3

2x + 3

€

y =3

2x + 3

Neither Coinciding

Write an equation of the line that passes through the points given:

€

m =y2 − y1

x2 − x1

=−8

8= −1

€

y − y1 = m(x − x1)

y − 4 = −1(x + 2)

y − 4 = −x − 2

y = −x + 2

(-2,4) and (6,-4) (3,-5) and (0,4)

€

m =y2 − y1

x2 − x1

=9

−3= −3

€

y − y1 = m(x − x1)

y + 5 = −3(x − 3)

y + 5 = −3x + 9

y = −3x + 4

Write an equation of a line using the information given.

1. No slope, (3,4) 2. slope = 3, (-3, -7)

€

y − y1 = m(x − x1)

y + 7 = 3(x + 3)

y + 7 = 3x + 9

y = 3x + 2

Slope is undefinedVERTICAL LINE

€

x = 3

How can you tell if two lines are perpendicular? Their slopes are opposite reciprocals

HOW CAN WE TELL IF THEY ARE PARALLEL?

Their slopes are the SAME

Given f(x) and g(x), find (f/g)(x)

€

f (x) = 2x 2 − 3x

g(x) = x − 5

€

f (x) = −4x 2 − 3x +10

g(x) = 6x −1

€

2x 2 − 3x

x − 5,x ≠ 5

€

−4x 2 − 3x +10

6x −1,x ≠

1

6

Solve this system of three variables:

Find the product of each:

€

1 −3

0 4

⎡

⎣ ⎢

⎤

⎦ ⎥•

1 5 −2

0 4 0

⎡

⎣ ⎢

⎤

⎦ ⎥

€

1 5 −2

0 4 0

⎡

⎣ ⎢

⎤

⎦ ⎥•

1 −3

0 4

⎡

⎣ ⎢

⎤

⎦ ⎥

DOES NOT EXIST

€

1 −7 −2

0 16 0

⎡

⎣ ⎢

⎤

⎦ ⎥

2X3 2X2

Evaluate the determinant of this 3x3

matrix:

€

1 −2 4

3 0 4

−7 1 3

€

3 −4 0

1 3 7

−10 0 2

1 -2

3 0

-7 1

DOWNHILL - UPHILL

(0+56+12) - (0+4-18)

68 – (-14)

82

(18+280+0) - (0+0-8)

3 -4

1 3

-10 0

246+8

254

Evaluate each function given:1. f(a2) 2. f(3b4)

€

f (x) = 2x 2 − 3x + 2

€

2(a2)2 − 3(a2) + 2

2a4 − 3a2 + 2

€

2(3b4 )2 − 3(3b4 ) + 2

18b8 − 9b4 + 2

Graph each function:1. f(x) = 3x – 4 2. f(x) = -⅔x + 1

Find the values of x and y for which the matrix equation is

true.

€

x − y x[ ] = 1 3 − y[ ]

€

3x − 2y y[ ] = 15 −3x + 6[ ]

€

x − y =1

x = 3 − y

I would use substitution:

€

(3 − y) − y =1

3 − 2y =1

−2y = −2

y =1

€

x = 3 − (1)

x = 2

(2,1)

€

3x − 2y =15

y = −3x + 6

I would use substitution:

€

3x − 2(−3x + 6) =15

3x + 6x −12 =15

9x = 27

x = 3

€

y = −3(3) + 6

y = −3

(3,−3)

Given the two matrices, perform the following operations.

A = B =

€

1 6 −1

0 3 −2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

1 −4 4

11 0 50

−2 0 −1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

1. 3B 2. 2A - C

€

3 18 −3

0 9 −6

⎡

⎣ ⎢

⎤

⎦ ⎥ Impossible

Find the inverse of each matrix.

1. 2.

€

−1 3

4 7

⎡

⎣ ⎢

⎤

⎦ ⎥

€

−2 3

4 −6

⎡

⎣ ⎢

⎤

⎦ ⎥

€

1

−19

7 −3

−4 −1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

−719

319

419

119

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

1

0

−6 −3

−4 −2

⎡

⎣ ⎢

⎤

⎦ ⎥

Does not exist

Graph each inequality:

1. 2x + y – 3 < 0 2. x + 3y – 6 ≥ 0

Determine the intervals of increasing and decreasing for

each function:

€

f (x) = x 2 − 2x +1

€

f (x) = x 3 + 2x 2 − x + 4

Decreasing x < 1Increasing x > 1

Decreasing -1.5 < x < 0.2Increasing x < -1.5, x > 0.2

What lines are symmetric to each

function given:1. 2.

€

x 2

4+y 2

9=1

x = 0

y = 0

€

(x − 4)2

4+

(y + 2)2

9=1

x = 4

y = -2

Graph each function and it’s inverse.

1. 2.

€

f (x) = x 2 + 3

€

f (x) = x − 2

f(x)

f-1(x)

f(x)

f-1(x)

Determine whether the critical pt given is a max, min, or pt of

inflection.

1.

€

f (x) = 3x 3 −18x 2 − 4 x = 0 2.

€

f (x) = 3x 3 − 9x + 5 x = 1

€

(−0.1,−4.183)

(0,−4)

(0.1,−4.177)

MAX€

(0.9,−.913)

(1,−1)

(1.1,−.907)

MIN

Approximate the real zero.

1. 2.

€

f (x) = x 3 + 2x 2 − 3x − 5

€

f (x) = x 4 − 8x 2 +10

x y

-5 -65-4 -25-3 -5-2 1-1 -10 -51 -52 53 31

x y

-5 435-4 138-3 19-2 -6-1 30 101 32 -63

19

So there is zeroes between -3 and -2, -2 and -1, 1 and 2

So there is zeroes between -3 and -2, -2 and -1, 1 and 2

Rule of thumb: go from -5 to 5 for your x-values

If they want a decimal approximation, you need to make another t-chart going by 0.1 in between these approximated zeros.

Or you could just plug each answer and see which one gets you closest to a ZERO

Solve the system of inequalities by graphing

€

x > −2

y > 0

x + y < 3

3x - y < 2

Use the related function to find the min and max.1. 2.

€

f (x,y) = 3x + 2y

(2,3)(−1,8)(0,5)

€

l(x,y) = 35x − 20y +10

(−3,3)(−1,1)(0,−2)

Determine the vertical asymptotes of each

function

€

f (x) =x

5x

€

f (x) =x + 2

3x −1

€

f (x) =2x − 5

x 2 − 4x

VA: x = 0 VA: x = ⅓

VA: x = 4, x = 0

Graph each rational function

€

f (x) =x 2 − 4

x + 2

€

f (x) =x 2 + 5x

x

€

(x + 2)(x − 2)

x + 2

€

x(x + 5)

x

Hole at x = -2

Hole at x = 0

Find the roots of:

€

x 3 + x 2 −11x +10 = 0

A.) B.) C.) 2, -1 D.) -2, 1

€

2,−3 ± 29

2

€

2,3 ± i 29

2

USE THE COMMON ROOT AND DO SYNTHETIC DIVISION FIRST

2 IS COMMON AMONG ALL THE ANSWERS

AFTER SYNTHETIC DIVISION,TRY TO FACTOR, OR QUADRATIC FORMULATO FIND THE REST OF THE ROOTS.

Find the number of positive, negative, and imaginary roots

possible for this function:

€

f (x) = 2x 5 − x 4 + 2x 3 + x −10 3, 1 positive roots

€

f (−x) = −2x 5 − x 4 − 2x 3 − x −10 0 Negative roots

P N I

3 0 2

1 0 4

Each row adds up to degree of polynomial

In a polynomial equation, if there is four changes in signs of the coefficients of the terms, __________________________there is 3 or 1 positive roots

Using Law of Sines1. In ΔABC if A = 63.17°, b = 18, and a = 17, find B

2. In ΔABC if A = 29.17°, B = 62.3°, and c = 11.5, find a

Determine the type of discontinuity for each function:

Find the maximum value for this system of inequatilites:

Infeasible? Unbounded? Optimal solutions?

Solve this rational inequality:

Use a number line

Find this trig value:

1. Given

Evaluate each problems using the unit circle:

€

tanπ

4=

tan2π

3=

tan(−150°) =

€

1

− 3

3

3

Determine for each function if it is odd, even, or neither?

€

y = x

x 2 + y 2 = 9

y = x 3

y = x 2

Odd functions are symmetric with respect to the origin:

(a,b) and (-a,-b)

Even functions are symmetric with respect to the y-axis:

(a,b) and (-a,b)

EVEN

BOTH

ORIGIN

EVEN

List all possible rational roots of each function:

€

x 3 − 2x 2 + 3x −10

€

4x 3 − x 2 + 5x + 3

P: 1, 2, 5, 10Q: 1

€

±1,±2,±5,±10

P: 1,3Q: 1, 2, 4

€

±1,±3,±1

2,±

3

2,±

1

4,±

3

4

Use the triangles below to find missing cos A, sin A, tan A

A

8 ft.

5 ft.

€

cosA =

sinA =

tanA =€

89

€

8 89

89

€

5 89

89

€

5

8

Use the unit circle to find each:

€

tan180° =

sec270° =

sin5π

4=

csc(−90°) =

0

undefined

€

− 2

2

-1

State the amplitude for each function:

€

y = tan θ − 45°( )

€

y = 2sin 3θ −π

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

y = secθ

3−π

2

⎛

⎝ ⎜

⎞

⎠ ⎟+ 3

Amplitude = none Amplitude = 2

Amplitude = 1

Find the period for each function:

€

y = tan θ − 45°( )

€

y = 2sin 3θ −π

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

y = secθ

3−π

2

⎛

⎝ ⎜

⎞

⎠ ⎟+ 3

Period = π/k = π Period = 2π/k = 2π/3 or 120°

Period = 2π/k = 6π or 1080°

Graph each function

€

f (x) =1

x + 3

€

f (x) =1

x − 5

VA: x = -3HA: y = 0

VA: x = 5HA: y = 0