is the graph a function or a relation? function relation
TRANSCRIPT
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Pre-Calculus Midterm Exam Review
I’m excited!
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Is the graph a function or a relation?
Function Function
Relation
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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
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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
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Find the x- and y- intercepts:
€
x + 2y −12 = 0
€
−4x + 6y + 24 = 0
(12,0) and (0,6) (6,0) and (0,-4)
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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
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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
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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
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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
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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
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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
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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
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Solve this system of three variables:
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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
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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
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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
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Graph each function:1. f(x) = 3x – 4 2. f(x) = -⅔x + 1
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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)
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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
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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
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Graph each inequality:
1. 2x + y – 3 < 0 2. x + 3y – 6 ≥ 0
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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
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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
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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)
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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
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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
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Solve the system of inequalities by graphing
€
x > −2
y > 0
x + y < 3
3x - y < 2
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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)
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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
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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
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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.
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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
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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
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Determine the type of discontinuity for each function:
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Find the maximum value for this system of inequatilites:
Infeasible? Unbounded? Optimal solutions?
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Solve this rational inequality:
Use a number line
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Find this trig value:
1. Given
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Evaluate each problems using the unit circle:
€
tanπ
4=
tan2π
3=
tan(−150°) =
€
1
− 3
3
3
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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
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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
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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
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Use the unit circle to find each:
€
tan180° =
sec270° =
sin5π
4=
csc(−90°) =
0
undefined
€
− 2
2
-1
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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
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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°
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Graph each function
€
f (x) =1
x + 3
€
f (x) =1
x − 5
VA: x = -3HA: y = 0
VA: x = 5HA: y = 0