Algebra I Flashcards
What is a Geometric Sequence?
- 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
Common Ratio
- 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
Geometric Series
- 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
Function
- 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
Domain
- the range of inputs for a function
- the x axis inputs
Range
- the range of outputs for a function
- the y axis outputs
What is the basic Parabola Vertex Formula?
Quadratic Equation
Axis of Symmetry
- the line that goes through the vertex of a parabola and that the parabola equally reflects around.
Two ways to find the Vertex of a Parabola
- 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
What are the Roots of a Function?
- points where the x axis is intersected
- these are the point(s) where y is equal to zero
How do you find the Roots of a Function?
- x2-11x+30
- (x-6)(x-5)
- Roots: x = 6 , x = 5
What is the Vertex of a Parabola?
- the lowest or highest y value of a parabola
- lowest = positive parabola
- highest = negative parabola
What is the Standard Form?
- form of an equation with
- variables and coefficients on one side
- constants on the other
- integers only
- ax + by = c
- -14x + 21y = 12
How does an exponent operation work?
- 32 = 1 * 3 * 3 = 9
- 02 = 1 * 0 * 0 = 0
- 50 = 1
- Also, growth^duration = new/original
- 32 = 9/1 = 9
Point-Slope Form
Slope-Intercept Form
y = mx + b
Negative Exponent
- invert the base number then perform the exponent operation
Why does exponentation start with 1?
- 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)
Why does 00 = 1?
- 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
What is a good visual analogy for exponentation?
- a tree!
How do you factor a quadratic by grouping?
- 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)
What is a rational number?
- 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…
What are the rules for rational numbers?
- 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
How do you factor a quadratic that doesn’t share factors?
- 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)
Do both the addition and subtraction rules apply to fractional exponents?
- yes!
What is the equation for the vertext of a parabola from a quadratic?
- y = a(x-h)2 + k
- (h,k) = vertex
- a = coefficient of x2
What element of the quadratic tells you whether a parabola opens up or opens down?
- the coefficient of x2
- “a” in ax2+bx+c
- positive “a” opens upwards and negative “-a”opens downwards
Explain how the parabola vertex equation works
- 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
Convert a quadratic to the equation of the vertex of a parabola
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)
How do you solve the for a linear equation to figure out upward or downward shading?
- 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
How do you figure out the amount of solutions for an equality of two equations?
- 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)
How do you find the domain of a function involving a square root?
- it must take the square root of a non-negative number
- f(x) = sqrt(2x-8)
- 2x-8 >= 0
- 2x >= 8
- x >= 4
How do you find the domain of a function that is a fraction?
- the denominator must not equal zero != 0
- f(x) = (-4-5t)/(1-2t)
- 1-2t != 0
- -2t != -1
- t != 1/2
What is a piecewise funcion?
- a function that has different range equations for different number values
Give examples illustrating operators and bracket types for piecewise equations.
(-6, 0] = -6 < x <= 0
[-1, 22] = -1 <= x <= 22
(5, 9) = 5 < x < 9
What is the form of a perfect square?
(A+B)2 = Ax2+2ABx+B2
How do you find and factor a perfect square?
- look for perfect square nomials
- see if they fit the form Ax2+2AB+B2
- if they do then they should equal (A+B)2
What is Newton’s Formula for finding the square root?
- 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
- …
What is the absolute value of a number?
- 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
How do you interpret an absolute value equation?
- 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
How do you graph an absolute value?
- 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
- an absolute value that equals zero
- |x+9| = 0
- x = -9
- any value other than zero will have two solutions
How do you solve an absolute value equality?
- 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
How do you solve a function that has a square root?
- solve for the square root so that the answer is greater than or equal to zero: sqrt >= 0
William is twice as old as Kevin now. 10 years ago William was 7 times as old as Kevin. How old is William now?
- 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)
How do you solve for b in the slope intercept form with two known points? (x,y) (x1,y1)
- 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
What is the discriminant of the quadratic equation?
- it is the value inside the square root portion of the numerator: b2-4ac
How do you determine the amount of solutions to a quadratic?
- 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
How do you determine whether the discriminant of a quadratic is positive, negative or zero by looking at the graph?
- 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
How do you find out if monomials with an integer coefficient (ex. ?4x2y4) is a factor of another?
- 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
How do you solve for a function f(v) = u, in terms of v with the following: u-5 = -4(v-1)?
-
solve the equation for u
- u-5 = -4(v-1)
- u-5 = -4v+4
- u = -4v+4+5
- u = -4v+9
- -4v + 9
How do you properly convert a word problem to a function formula?
- 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
In a word problem, what is “rate” equivalent to?
- slope (m); any stated rate essentially tells you the slope
What is the easiest way to find b in the point-slope form?
- the y value when x = 0; so in (0, 2.9), b =2.9
How do you solve a word problem with a system of equations?
- 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
What is a system of linear equations?
- two or more linear equations that each have the same unknowns
- it can be solved through substitution, subtraction, graphing, and matrices
In what visual ways can you solve a structure word problem?
- visualize the breakdown of each element
In what algebraic ways can you solve a structure word problem?
- 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
How do you evaluate sequences in recursive form?
- 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
How do you factor polynomials with non-zero leading coefficients?
- complete the square
- find squares in the first and last terms
- EXAMPLE
- 36k2+12k+1
- 36 = 62; 1 = 12
- (6k+1)2
What is an arithmetic sequence?
- 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
What is the common difference between terms?
- 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
What is a recursive equation for a geometric sequence?
for g(n)
500 if n = 1
g(n-1) * 1/5 if n > 1
What is the easiest form of a quadratic to reveal the y-intercept?
- standard form: Ax2+Bx+C
- C = y-intercept
- 3x2+36x+33
- 3(0)2+36(0)+33
- 0+0+33
- 33
What form of a quadratic most easily reveals the “zeros” of the equation?
- 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}
What is the form of a quadratic that most easily reveals the vertex?
- vertex form: a(x-h)2+k
- vertex= (h,k)
- 2(x-8)+24
- vertex = (8,24)
What is the formula for x of the vertex of a parabola from the standard form?
standard form = ax2+bx+c
x = -b/2a
What is the best method to use to interpret quadratic models?
- 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)22+180P
What is the difference between whole numbers and integers?
- 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
What is another way to define or think about the average rate of change?
- slope
- m = (y1-y2)/(x1-x2)
- slope-intercept form: y = mx+b
- starting value = b
- plug in the values to the equation and either
- find the slope
- create a table and find the rate of change from that
How do you find the average rate of change from a table?
- simply put the function change over the input change
What are the relative minimum and maximum points? How do you express them mathematically?
- 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
What are the absolute minimum and maxiumum points?
- the absolute maxiumum and minimum points on a graph over the entire graph
What is an exponential function?
- 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
What is the initial value of an exponential function?
- 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
- 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)
- f(x) = 1/5 * 3x
What is the form of an exponential equation?
g(x) = a * r2
How do you solve an exponential equation with a graph or series of points?
- 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
- 4 = a * r0 -
- g(x) = 4 * 2x
In a g(x) exponential equation how do you solve for x when the common ratio and initial value are given?
- 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
What are the best points to find when solving an exponential equation?
- find the y-intercept to easily find a
- for (0,8)
- 8 = a * r0
- 8 = a * 1
- a = 8
- for (0,8)
- 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
- for (1,5)
What is the initial value of an exponential equation on the graph?
- the y value of the y-intercept
How do you solve complex, mult-degree equations?
- 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
What is the simplest way to solve a quadratic equation where the leading coefficient is != 1?
- 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
What is the most elegant way to solve complex quadratics?
- 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
What is the simple solution to remember when solving a complex quadratic?
- use the quadratic formula!
How do you solve for r in an exponential equation?
- 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
How do you solve for a of an exponential function?
- 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’
How do you use graph values to solve an exponential equation?
- 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
What is a stem and leaf plot?
- 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
