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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:. All real numbers except 1 or -1. All real numbers except 3 or -3. All real numbers except 5. All real numbers except 0 and 5. - PowerPoint PPT Presentation
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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 −1x 2 − 9
€
y =xx − 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 − 2g(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
€
23
€
0 = −12x 2 − 48
48 = −12x 2
−4 = x 2
x = −4x = ±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 =12x − 6
€
y = −4x + 20
€
y =32x + 3
€
y =32x + 3
Neither Coinciding
Write an equation of the line that passes through the points given:
€
m =y2 − y1
x2 − x1
=−88
= −1
€
y − y1 = m(x − x1)y − 4 = −1(x + 2)y − 4 = −x − 2y = −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 + 9y = −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 + 9y = 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 − 3xg(x) = x − 5
€
f (x) = −4x 2 − 3x +10g(x) = 6x −1
€
2x 2 − 3xx − 5
,x ≠ 5
€
−4x 2 − 3x +106x −1
,x ≠16
Solve this system of three variables:
Find the product of each:
€
1 −30 4 ⎡ ⎣ ⎢
⎤ ⎦ ⎥•
1 5 −20 4 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
1 5 −20 4 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥•
1 −30 4 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
DOES NOT EXIST
€
1 −7 −20 16 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
2X3 2X2
Evaluate the determinant of this 3x3
matrix:
€
1 −2 43 0 4
−7 1 3
€
3 −4 01 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 =1x = 3 − y
I would use substitution:
€
(3 − y) − y =13 − 2y =1−2y = −2y =1
€
x = 3 − (1)x = 2
(2,1)
€
3x − 2y =15y = −3x + 6
I would use substitution:
€
3x − 2(−3x + 6) =153x + 6x −12 =159x = 27x = 3
€
y = −3(3) + 6y = −3
(3,−3)
Given the two matrices, perform the following operations.
A = B =
€
1 6 −10 3 −2 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
1 −4 411 0 50−2 0 −1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
1. 3B 2. 2A - C
€
3 18 −30 9 −6 ⎡ ⎣ ⎢
⎤ ⎦ ⎥ Impossible
Find the inverse of each matrix.
1. 2.
€
−1 34 7 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
−2 34 −6 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
1−19
7 −3−4 −1 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
−719
319
419
119
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ €
10
−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 > −2y > 0x + y < 33x - 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 + 23x −1
€
f (x) =2x − 5x 2 − 4x
VA: x = 0 VA: x = ⅓
VA: x = 4, x = 0
Graph each rational function
€
f (x) =x 2 − 4x + 2
€
f (x) =x 2 + 5xx
€
(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:
€
tanπ4
=
tan2π3
=
tan(−150°) =
€
1
− 3
33
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,±12
,±32
,±14
,±34
Use the triangles below to find missing cos A, sin A, tan A
A
8 ft.
5 ft.
€
cosA =
sinA =
tanA =€
89
€
8 8989
€
5 8989
€
58
Use the unit circle to find each:
€
tan180° =
sec270° =
sin5π4
=
csc(−90°) =
0
undefined
€
− 22
-1
State the amplitude for each function:
€
y = tan θ − 45°( )
€
y = 2sin 3θ −π4
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
y = secθ3
−π2
⎛ ⎝ ⎜
⎞ ⎠ ⎟+ 3
Amplitude = none Amplitude = 2
Amplitude = 1
Find the period for each function:
€
y = tan θ − 45°( )
€
y = 2sin 3θ −π4
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
y = secθ3
−π2
⎛ ⎝ ⎜
⎞ ⎠ ⎟+ 3
Period = π/k = π Period = 2π/k = 2π/3 or 120°
Period = 2π/k = 6π or 1080°
Graph each function
€
f (x) =1x + 3
€
f (x) =1x − 5
VA: x = -3HA: y = 0
VA: x = 5HA: y = 0