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Math > Algebra I > Flashcards

Algebra I Flashcards

1
Q

What is a Geometric Sequence?

A
  • a sequence with a constant increase or decrease involving multiplication or division
  • each term is found by multiplying the previous term by a constant
    • 2, 4, 8, 16, 32, 64, 128, 256, …
  • z(x)n-1
    • z=base
    • x=common ratio
    • n=sequence number
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2
Q

Common Ratio

A
  • A ratio of a term to the previous or subsequent term in a geometric sequence
  • in {2, 4, 8, 16, 32, 64, 128, 256, …} the common ratio is 2
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3
Q

Geometric Series

A
  • the sum of a geometric sequence
  • 2(3)1-1 + 2(3)2-1+2(3)3-1+2(3)4-1+2(3)5-1
  • 2 + 6 + 18 + 54 + 162
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4
Q

Function

A
  • a function takes one input (x) and generates one specific output (y)
  • f(x) = y
  • it is NOTa function if there can be multiple outputs for one input
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5
Q

Domain

A
  • the range of inputs for a function
  • the x axis inputs
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6
Q

Range

A
  • the range of outputs for a function
  • the y axis outputs
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7
Q

What is the basic Parabola Vertex Formula?

A
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8
Q

Quadratic Equation

A
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9
Q

Axis of Symmetry

A
  • the line that goes through the vertex of a parabola and that the parabola equally reflects around.
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10
Q

Two ways to find the Vertex of a Parabola

A
  • create a perfect square from the quadratic
  • input an x value so that the perfect square equals zero
  • this is the x value of the vertex
  • the solution to the equation is the y value of the vertex

Alternate

  • if you have the two roots of the parabola…
  • take the average of the two roots
  • then plug that number into the quadratic equation
  • that solution and the average x value equals the vertex
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11
Q

What are the Roots of a Function?

A
  • points where the x axis is intersected​
  • these are the point(s) where y is equal to zero
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12
Q

How do you find the Roots of a Function?

A
  • x2-11x+30
  • (x-6)(x-5)
  • Roots: x = 6 , x = 5
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13
Q

What is the Vertex of a Parabola?

A
  • the lowest or highest y value of a parabola
  • lowest = positive parabola
  • highest = negative parabola
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14
Q

What is the Standard Form?

A
  • form of an equation with
    • variables and coefficients on one side
    • constants on the other
    • integers only
  • ax + by = c
  • -14x + 21y = 12
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15
Q

How does an exponent operation work?

A
  • 32 = 1 * 3 * 3 = 9
  • 02 = 1 * 0 * 0 = 0
  • 50 = 1
  • Also, growth^duration = new/original
  • 32 = 9/1 = 9
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16
Q

Point-Slope Form

A
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17
Q

Slope-Intercept Form

A

y = mx + b

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18
Q

Negative Exponent

A
  • invert the base number then perform the exponent operation
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19
Q

Why does exponentation start with 1?

A
  • exponentation, and arithmetic in general, is the transformation of numbers
  • think of exponentation as starting with a scaling factor, 1 then trasforming it
  • start with the scaling factor: 1
  • you set the growth factor per interval, x in xn​ = y
  • you set the amount of intervals, n in xn​ = y
  • the answer is the transformed number, y in xn = y
  • 34 = 1 (base) * 3 (growth, interval 1) * 3 (interval 2) * 3 (interval 3) * 3 (interval 4) = 81 (transformed number)
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20
Q

Why does 00 = 1?

A
  • use the microwave analogy
  • in 01 think of the microwave as activating the growth factor, in this case 0, so the 0 turns on and obliterates the 1 through multiplication
  • however in 00 the growth factor is activated 0 times, it never turns on, so all that’s left is the scaling factor, 1
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21
Q

What is a good visual analogy for exponentation?

A
  • a tree!
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22
Q

How do you factor a quadratic by grouping?

A
  • y = Ax2 + Bx + C
  • ab = A*C
  • a+b = B
  • find the factors that satisfy these two equations
  • group then factor out terms that make the equations in parenthesis equal
  • 4x2+25x-21
    • 4 * -21 = -84
    • -3 * 28 = -84
    • -3 + 28 = 25
    • 4x2+28x -3x-21
    • 4x(x+7) * -3(x+7)
    • (x+7) * (4x-3)
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23
Q

What is a rational number?

A
  • a number that can be expressed as the ratio of two integers
    • 9 = 9/1, 27/3
    • 3/2, 7/8
  • decimals that terminate or repeat are rational
    • 0.999…
    • 0.425425…
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24
Q

What are the rules for rational numbers?

A
  • the product of a non-zero rational number and an irrational number is irrational: r * i = r
  • the reciprocal of an irrational number is irrational: 1/i = i
  • the sum of a rational and irrational number is irrational: i + r = i
  • the sum or product of two irrational numbers could be rational or irrtational: i + i OR i * i = i OR r
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25
Q

How do you factor a quadratic that doesn’t share factors?

A
  • find a and b, where a+b = B and a*b = A*C
  • Ex: 8x2-18x-5
    • a+b = B = -18
    • a*b = A*C = -40
    • a = 2, b = -20
  • Rewrite as Ax2+ax+bx+C, and group
    • 8x2+2x-20x-5
    • (8x2+2x)+(-20x-5) = 2x(4x+1)+-5(4x+1)
    • (2x-5)(4x+1)
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26
Q

Do both the addition and subtraction rules apply to fractional exponents?

A
  • yes!
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27
Q

What is the equation for the vertext of a parabola from a quadratic?

A
  • y = a(x-h)2 + k
  • (h,k) = vertex
  • a = coefficient of x2
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28
Q

What element of the quadratic tells you whether a parabola opens up or opens down?

A
  • the coefficient of x2
  • “a” in ax2+bx+c
  • positive “a” opens upwards and negative “-a”opens downwards
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29
Q

Explain how the parabola vertex equation works

A
  • y = a(x-h)2 + k
  • (h,k) = vertex
  • a = coefficient of x2
  • if “a” == positive, parabola opens upwards
  • if “a” == negative, parabola opens downwards
  • if “a” == positive AND x = h
    • a(x-h)2 = 0, therefore a(x-h)2 + k = k
    • k is the lowest point of y, the vertex
    • h is the x value of the vertex
  • if “a” == negative AND x = h, k is the highest point of y, the vertex
    • a(x-h)2 = 0, therefore a(x-h)2 + k = k
    • k is the highest point of y, the vertex
    • h is the x value of the vertex
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30
Q

Convert a quadratic to the equation of the vertex of a parabola

A

vertex equation: y = a(x-h)2+k

vertex: (h,k)

y= 2x2-8x+20

2(x2-4x+10)

2(x2-4x+4)+10-4

2(x-2)2+6

vertex: (2,6)

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31
Q

How do you solve the for a linear equation to figure out upward or downward shading?

A
  • convert to slope-intercept form: y = mx + b
  • 2x + 2y > 6
    • 2y > -2x + 6
    • y > -x + 3
    • so the shaded area is ABOVE the dashed line
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32
Q

How do you figure out the amount of solutions for an equality of two equations?

A
  • resolve both sides: 5x-9x+3 = -4x+6-3
    • -4x+3 = -4x+3
    • 3=3 (infinite solutions)
  • solve for x: 10x+7-1 = 5x+4
    • 10x-5x=4-6
    • 5x=-2
    • x=-2/5 (one solution)
  • remove constants or variables: 11x-6 = 15x+7-4x
    • 11x-6 = 11x+7
    • -6=7 (no solutions)
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33
Q

How do you find the domain of a function involving a square root?

A
  • it must take the square root of a non-negative number
  • f(x) = sqrt(2x-8)
    • 2x-8 >= 0
    • 2x >= 8
    • x >= 4
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34
Q

How do you find the domain of a function that is a fraction?

A
  • the denominator must not equal zero != 0
  • f(x) = (-4-5t)/(1-2t)
    • 1-2t != 0
    • -2t != -1
    • t != 1/2
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35
Q

What is a piecewise funcion?

A
  • a function that has different range equations for different number values
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36
Q

Give examples illustrating operators and bracket types for piecewise equations.

A

(-6, 0] = -6 < x <= 0

[-1, 22] = -1 <= x <= 22

(5, 9) = 5 < x < 9

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37
Q

What is the form of a perfect square?

A

(A+B)2 = Ax2+2ABx+B2

38
Q

How do you find and factor a perfect square?

A
  • look for perfect square nomials
  • see if they fit the form Ax2+2AB+B2
  • if they do then they should equal (A+B)2
39
Q

What is Newton’s Formula for finding the square root?

A
  • first give an educated guess at the answer
  • then apply the following formula repeatedly to get closer to the actual answer
  • 1/2(N / A + A) = New Answer (NA)
  • 1/2(N/NA + NA) = NA
  • …
40
Q

What is the absolute value of a number?

A
  • it is the distance of that number from zero
  • both |-5| and |5| are 5 away from zero
  • think of the number line and zero as the mid point
41
Q

How do you interpret an absolute value equation?

A
  • think of it as how far x is away from the constant
  • ex: in |x-5| = 10 you’re asking what numbers are 10 places away from the absolute value of -5 |-5| on the number line
    • -5 is 10 away from 5
    • 15 is 10 away from 5
  • Or get the absolute value of the answer: |x-5| = 10 is x-5 = 10 OR x-5 = -10
42
Q

How do you graph an absolute value?

A
  • y = |x+3|
  • because the full equation is an absolute value it can’t cross the x-axis and will graph as a V on the x-axis
  • an equation with a value outside of the absolute value can pass the x-axis: y = |x+3| - 10
43
Q
A
  • an absolute value that equals zero
    • |x+9| = 0
    • x = -9
  • any value other than zero will have two solutions
44
Q

How do you solve an absolute value equality?

A
  • get the absolute value on one side and the constants on the other
  • eliminate coefficients through division
  • solve for the absolute value normally, with the + and - value of the constant
  • EXAMPLE
    • 2|x+10|+12 = -3|x+10|+62
    • 5|x+10| = 50
    • |x+10| = 10
    • x+10 = +-10
    • x = 0, x = -20
45
Q

How do you solve a function that has a square root?

A
  • solve for the square root so that the answer is greater than or equal to zero: sqrt >= 0
46
Q

William is twice as old as Kevin now. 10 years ago William was 7 times as old as Kevin. How old is William now?

A
  • w = 2k
  • w-10 = 7(k-10)
  • w-10 = 7k-70
  • k = 12
  • William is now 24 and Kevin is 12. 10 years ago William was 14 and Kevin was 2 (2*7=14)
47
Q

How do you solve for b in the slope intercept form with two known points? (x,y) (x1,y1)

A
  • y = mx + b
  • find the slope using the slope formula m = y-y1/x-x1
  • plug the slope into the slope intercept equation along with the x,y of one of the points to find b
  • EXAMPLE
    • (2,1) (4,2)
    • m = (1-2)/(2-4) = -1/-2 = 1/2
    • 1 = 1/2(2) + b
    • 1 = 1 + b
    • b = 0
48
Q

What is the discriminant of the quadratic equation?

A
  • it is the value inside the square root portion of the numerator: b2-4ac
49
Q

How do you determine the amount of solutions to a quadratic?

A
  • use discriminant of the quadratic equation
  • if b2-4ac > 0, then there will be two solutions
  • if b2-4ac = 0, then there will be one solution
  • if b2-4ac < 0, then there will be no solutions
50
Q

How do you determine whether the discriminant of a quadratic is positive, negative or zero by looking at the graph?

A
  • a parabola that intersects the x-axis at two points (two roots) has a positive discriminant
  • a parabola that doesn’t the x-axis (no roots) has a negative discriminant
  • a parabola that touches the x-axis at one point (repeated root) has a zero discriminant​
51
Q

How do you find out if monomials with an integer coefficient (ex. ?4x2y4) is a factor of another?

A
  • divide one by the other, and if you get a monomial with an integer coefficient as the quotient then it is “divisible”
  • EXAMPLE
    • is 2xy2 a factor of 4x2y4?
    • 4x2y4/2xy2 = 2xy2 = YES
52
Q

How do you solve for a function f(v) = u, in terms of v with the following: u-5 = -4(v-1)?

A
  • solve the equation for u
    • u-5 = -4(v-1)
    • u-5 = -4v+4
    • u = -4v+4+5
    • u = -4v+9
    • -4v + 9
53
Q

How do you properly convert a word problem to a function formula?

A
  • take the information in the problem and convert to slope-intercept form: y = mx + b
  • EXAMPLE
    • A lake near the Arctic Circle is covered by a 2-meter-thick sheet of ice. In spring the warm air melts the ice and after 3 weeks it’s 1.25 meters thick.
    • Let S(t) denot the ice sheet’s thickness S(measured in meters) and t represent time in weeks.
    • S(0) = m(0) + 2
      • b = 2
    • Two points representing the data points: (0,2) (3, 1.25)
      • m = 0.25
    • S(t) = -0.25t + 2 or S(t) = 2 - 0.25t
54
Q

In a word problem, what is “rate” equivalent to?

A
  • slope (m); any stated rate essentially tells you the slope
55
Q

What is the easiest way to find b in the point-slope form?

A
  • the y value when x = 0; so in (0, 2.9), b =2.9
56
Q

How do you solve a word problem with a system of equations?

A
  • visualize first as much as possible
  • fully analyze the constraints and create equations that fully encompass the two aspects of the question
  • find the “lines” and create equations that fully describe them
  • include the rates (kilo/h, mph) as factors in the equations and cancel them out
57
Q

What is a system of linear equations?

A
  • two or more linear equations that each have the same unknowns
  • it can be solved through substitution, subtraction, graphing, and matrices
58
Q

In what visual ways can you solve a structure word problem?

A
  • visualize the breakdown of each element
59
Q

In what algebraic ways can you solve a structure word problem?

A
  • make the factors equivalent to the amount it is being compared to
  • in the problem if c>d>d, is b?c+d+b > 1/3
  • b -> b/3b = 1/3 > b/c+d+b because b is less than c & d
60
Q

How do you evaluate sequences in recursive form?

A
  • you’re evaluating a function that defines a recursive sequence
  • you start with a base case and subsequent terms build upon this base case, and in fact require them to find the answer
61
Q

How do you factor polynomials with non-zero leading coefficients?

A
  • complete the square
  • find squares in the first and last terms
    • EXAMPLE
    • 36k2+12k+1
      • 36 = 62; 1 = 12
      • (6k+1)2
62
Q

What is an arithmetic sequence?

A
  • a sequence where the difference between numbers is always the same, involving addition and subtraction
  • EXAMPLES
    • {1,5,9,13,17,21,25…} diff = +4
    • {28,19,10,1,-8…} diff = -9
    • {2,4,8,16,32…} X
      • not an arithmetic sequence, as the difference between numbers is different
63
Q

What is the common difference between terms?

A
  • the numerical difference between terms in a sequence
  • in {3,9,15,21,27} it is 6
  • in {5,-2,-9,-16} it is -7
64
Q

What is a recursive equation for a geometric sequence?

A

for g(n)

500 if n = 1

g(n-1) * 1/5 if n > 1

65
Q

What is the easiest form of a quadratic to reveal the y-intercept?

A
  • standard form: Ax2+Bx+C
    • C = y-intercept
  • 3x2+36x+33
    • 3(0)2+36(0)+33
    • 0+0+33
    • 33
66
Q

What form of a quadratic most easily reveals the “zeros” of the equation?

A
  • factored form: a(x-b)(x-c)
  • the zeros are the values for x that will make each factor equal zero
  • 3(x-3)(x+5)
    • zeroes: {3,-5}
67
Q

What is the form of a quadratic that most easily reveals the vertex?

A
  • vertex form: a(x-h)2+k
    • vertex= (h,k)
  • 2(x-8)+24
    • vertex = (8,24)
68
Q

What is the formula for x of the vertex of a parabola from the standard form?

A

standard form = ax2+bx+c

x = -b/2a

69
Q

What is the best method to use to interpret quadratic models?

A
  • use quadratic forms to find the vertex of the parabolas
  • ax2+bx+c -> a(x-h)+k (h,k)
  • The power in watts, P(c)P(c)P, left parenthesis, c, right parenthesis, that is generated by an electrical circuit depends on a current in amperes, ccc, and can be modeled using the function P(c)=-20(c-3)2+180P(c)=−20(c−3)​2​2+180P
70
Q

What is the difference between whole numbers and integers?

A
  • whole numbers are all natural numbers including: 0,1,2,3,4,etc
  • integers are all whole numbers and their negative counterparts: -3,-2,-1,0,1,2,3
71
Q

What is another way to define or think about the average rate of change?

A
  • slope
  • m = (y1-y2)/(x1-x2)
  • slope-intercept form: y = mx+b
  • starting value = b
72
Q
A
  • plug in the values to the equation and either
    • find the slope
    • create a table and find the rate of change from that
73
Q

How do you find the average rate of change from a table?

A
  • simply put the function change over the input change
74
Q

What are the relative minimum and maximum points? How do you express them mathematically?

A
  • the maxiumum and minimum y points on a graph for a particular interval
  • they are larger or smaller than the points around them, the peaks or troughs of the graph
75
Q

What are the absolute minimum and maxiumum points?

A
  • the absolute maxiumum and minimum points on a graph over the entire graph
76
Q

What is an exponential function?

A
  • a function where the x value in f(x) is an exponent within the function
  • negative exponents are less than 1 and as they grow more negative they get closer to 0 but never exceed it
  • after x = 1 the line quickly climbs steeply up the y axis
77
Q

What is the initial value of an exponential function?

A
  • the value before the exponential factor is applied, or 1 * that value, where x is 0
  • in f(x) = 2 * 5x, the initial value is 2
    • f(0) = 2 * 50 = 2 * 1 = 2
78
Q
A
  • it is the base of the factor containing the exponent
  • in f(x) = 1/5 * 3x , the common ratio is 3
  • this is because 3 is what the exponent factor is being multiplied by, recursively
    • f(x) = 1/5 * 3x
      • f(0) = 1/5 * 30 = 1/5 * 1
      • f(1) = 1/5 * 31 = 1/5 * 3 * h(0)
      • f(2) = 1/5 * 32 = 1/5 * 3 * h(1)
      • f(3) = 1/5 * 33 = 1/5 * 3 * h(2)
79
Q

What is the form of an exponential equation?

A

g(x) = a * r2

80
Q

How do you solve an exponential equation with a graph or series of points?

A
  • treat them like a system of equations and solve using substitution
  • p1 = (0,4) , p2 = (2,8)
    • 4 = a * r0 -
      • 4 = a * 1
      • a = 4
    • 8 = 2 * r2
      • 4 = r2
      • r = 2
  • g(x) = 4 * 2x
81
Q

In a g(x) exponential equation how do you solve for x when the common ratio and initial value are given?

A
  • solve like any other equation
  • when you get to the exponent x simply convert it to a normal x variable and solve
  • you should get some multiple of the common ratio which will tell you what x is
  • g(x) = -4
    • -4 = -16/49 * (7/2)x
    • 49/4 = (7/2)2
    • x = 2
82
Q

What are the best points to find when solving an exponential equation?

A
  • find the y-intercept to easily find a
    • for (0,8)
      • 8 = a * r0
      • 8 = a * 1
      • a = 8
  • find the point where x = 1 to easily find r once you have a
    • for (1,5)
      • 5 = 8 * r1
      • 5/8 = r1
      • r = 5/8
83
Q

What is the initial value of an exponential equation on the graph?

A
  • the y value of the y-intercept
84
Q

How do you solve complex, mult-degree equations?

A
  • first multiply, then factor, then solve
  • find at least one x value to solve the equation
    • (4x+1)2+9(4x+1) = -18
    • multiply then add common factors
      • 162+44x+10= -18
    • factor the result
      • (4x+10)(4x+1) = -18
    • find
85
Q

What is the simplest way to solve a quadratic equation where the leading coefficient is != 1?

A
  • factor out the common factor for the whole equation then factor the remainder
  • 6x2 - 120x + 600 = 0
    • divide by 6
  • 6(x2 - 20x + 100) = 0
  • 6(x - 10)(x - 10) = 0
    • x = 10
86
Q

What is the most elegant way to solve complex quadratics?

A
  • look for structure and substitute your p’s
  • (3x+5)​2+5(3x+5)+6=0
    • common factor = 3x+5
    • change this common factor into a variable: 3x+5 = p
  • p2+5p+6 = 0
    • solve for p
  • (p+3)(p+2) = 0
  • so p = -3 OR p = -2
    • subsitute for p
  • 3x+5 = -3 AND 3x+5 = -2
  • so x = -8/3 OR x = -7/3
87
Q

What is the simple solution to remember when solving a complex quadratic?

A
  • use the quadratic formula!
88
Q

How do you solve for r in an exponential equation?

A
  • first take the ratio of one solution of g(x) to another
    • g(2) = 144
    • g(4) = 324
    • 144/324 = 9/4
  • then use the ratio to form the equation (a*r4)/(a*r2)
    • ‘a’ will cancel out leaving r4/r2 = r2
  • then make the two solutions equivalent: r2 = 9/4 = 3/2
  • OR simply make a table of numbers with x as an integer increasing by 1. the common ratio between the y values is r
    • x must be an integer and should increase by 1 to make the division/multiplication operation easier
89
Q

How do you solve for a of an exponential function?

A
  • Method 1: knowing r and or x, solve for a
  • Method 2: knowing r, find x = 0 by dividing or multiplying g(x) by r
    • ​x must be an integer and should increase by 1 to make the operation easier
  • y of x = 0 is the initial value ‘a’
90
Q

How do you use graph values to solve an exponential equation?

A
  • make a table of values with x as an integer increasing by 1
  • the ratio of y1 to y2 is the common ratio r
  • find x = 0, and the y value is the initial value a
91
Q
A
92
Q

What is a stem and leaf plot?

A
  • a grouping of numerical data where the “stem” is the first part of a value and the “leaves” represent the second part of the value

Math (2 decks)

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