Math

Linear Systems

Modelling With Linear Equations:

– linear is a straight line, have an x/y intercept

– linear system is made of two or more lines, and the point of intersection is the solution to the system

– point of intersection is a single point that satisfies both equations

– breakeven problems:

> revenue = profit – cost, but when revenue = 0, that is breakeven point

> let x = how much is spent, let y = how much we made

– relative value reasoning problems:

> let x = first number, let y = second number

– mixture problems:

> let x = amount of money invested at certain percent, let y = other percent

– rate problems:

> speed = distance/time

> let x = time is car, let y = time is train

Graphing by Hand:

– convert equation into slope form, and plot y intercept

– use slope to find next point, plot, then connect

– extend line, put arrows on ends

– title, label axis, number graphs, consistent scales, state POI

– can also use table of values/x/y-intercept method

> x intercept is when y = 0, y intercept is when x = 0

Ways Two Lines Can Intersect:

– number solutions:

> 1—two lines have different slopes and y intercept has no impact

> infinite—when 2 lines are multiples of each other

> no solutions—when two lines are parallel, have same slope but different y-intercepts

Substitution:

– POI solution to a system, single point that satisfies both lines, x and y will be same for both lines

> x_{1} = x_{2} and y_{1} = y_{2}

– steps:

> number equations

> isolate x or y on either side (y is easier)

> set one equation as equal to the other (y_{1} = y_{2}), then solve for x

> sub in x, and solve y

> state POI

– example:

2x = 5y = 1 4x – 2y = 3

5y = -2x +1 2y = 4x – 3

y = -0.4x + 1 y = 2x – 1.5

-0.4x + 1 = 2x – 1.5

x = 0.7 y = -0.1

Elimination:

– infinite number of solution means 2 lines are the same

– goal is to eliminate one of the variables and then solve for the other, then sub in to find eliminated coordinate

– steps:

> label lines

> line up variables on one side, constants on the other

> identify which one to eliminate and multiply the lines until the coefficients for that variable are the same

> add or subtract 2 lines to eliminate the variable

> sub it into either equation to solve for the other variable

> state POI

Correlation and Regression:

– correlation is how strong a relationship is between 2 variables—weak, strong, positive, negative, none

– based on how close the points are on a scatter plot to line of best fit—represented as r

– correlation between 1 and -1 (1 is perfect positive, -1 perfect negative)

– correlation of 0 means there is horizontal line, and no relationship

– line of best fit doesn’t have to go through origin, but has to go through as many points as possible

> remaining points are equal on either side

Coordinate Geometry

Finding the Length of the Line:

d = √(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}

Equation of a Circle:

x^{2} + y^{2} = r^{2}

– r is radius or distance from centre of circle to edge or half diameter

– works when centre is (0,0)

Midpoint:

M = x_{1+}x_{2}, y_{1+}y_{2}

2 2

– median: a line segment that goes from the vertex of one point to the opposite side and crosses at the midpoint

– use midpoint to find equation of median of opposite side

Perpendicular Bisector:

– a line that crosses through the midpoint of another line segment at the 90 degree angle

– doesn’t necessarily go through vertex

– to find it, find opposite side’s perpendicular slope, and midpoint

– can’t find length because it’s not a line segment

– perpendicular bisector and median can be the same line in an isosceles/equilateral triangle

Centroid, Circumcentre and Orthocentre:

– altitude line is a line that goes from vertex to opposite side and forms a 90 degree angle, but does not go through midpoint, also known as height

– centroid is a point of intersection of two median lines

> can also be found by taking average of the x and y coordinates of all 3 vertices

– circumcentre is the point of intersection of 2 perpendicular bisectors

> can be on the outside of the triangle

– orthocentre is the point of intersection of two altitude likes

> to find altitude, find slopes of opposite side, then negative reciprocal, then sub in the vertex to get equation

Quadratic Relations

Quadratics:

– in quadratic relations, the 2^{nd} difference is constant

– u-shaped graph that opens up or down and has a degree of two (exponent of x is 2)

– variable rate of change, also known as parabola

– 2^{nd} difference is difference in the 1^{st} differences

– curve of best fit has same rules as line of best fit

Properties of Quadratics:

– vertex: is lowest to highest point on a parabola

> opens down = highest, opens up = lowest

– optimum value: y coordinate of vertex, can be maximum or minimum

– axis of symmetry: x coordinate vertex, can be found using average of zeros, or two points with same y coordinate

– 2^{nd} difference is positive—opens up, negative—opens down

– zeros: x intercepts of parabola

> can be one, two or none

– (h,k) represent vertex

Zeros:

– can find zeros by factoring, then use two cases (example: 2x(x-3): 2x is a case, and x-3 is a case)

> set each case equal to 0, then solve for x

Role of the Zeros:

– factored form: y = (x-s)(x-t), where s and t are zeros

– 2 zeros when s ≠ t

– 1 zero when s = t

– when in factored form, cannot sub in zeros for any reason

Sub-Unit: Factoring

Expansion:

– FOIL for binomials—(2x -1)(3x+2): first, outer, inner, last

– squaring binomials—square first term, square last term, multiple both terms by each other and by two

Factoring:

– ask: common factor? 1^{st} and 2^{nd} terms perfect squares? difference of squares? a = 1?

Common Factoring:

– factor out greatest common factor

– for good form, if first term is a negative, factor out a negative

– when multiplying like bases, add exponents

– always common factor where possible first, with all types of factoring

Sum and Product:

– when a = 1

– x^{2} + bx + c

– need two numbers with a sum of b and product of c

– write directly as (x-s)(x-t), where s and t are the two numbers

Decomposition:

– when a ≠ 1

– two numbers with a product of ac and a sum of b

– sub in for b-term (eg. 2x^{2} + 4x – 3x – 6, where 4x – 3x used to be x)

– common factor first two terms, then last two terms

– common factor entire equation

Difference of Squares:

– only has 2 terms, which both must be perfect squares, and must be subtracted from each other

– take square root of both then write as (x + s)(x – s)

– eg. 20a^{2 }– 180

> 20(a^{2 }– 9)

> 20(a + 3)(a – 3)

Perfect Squares:

– a^{2}x^{2} +/- 2abx + b^{2}

– a and c must be perfect squares and b term must be 2ac

– write as (ax – b)^{2}

Partial Factoring:

– used to find AOS and optimal value where there are no zeros and cannot be factored completely

– finds two coordinates with the same y coordinates

– coordinates written as: (0, c) and (-b/a, c)

– factor out the x and leave c alone, then set both cases as set to zero

> y = 2x^{2} + 8x + 5

> y = x(2x + 8) + 5

> first case is x, second is 2x + 8

> x = 0 and x = -4—these are two points, use average to find AOS

Completing the Square:

– to go from standard to vertex form

y = 2x^{2} – 5x +1

– factor out the coefficient of the x-squared term, and leave c alone

y = 2(x^{2} – 2.5x) + 1

– take half of factored b term, then square it—then add and subtract to keep equation the same

y = 2(x^{2} – 2.5x + 1.25^{2} – 1.25^{2}) + 1

y = 2(x^{2} – 2.5x + 1.56 – 1.56) + 1

– take out subtracted term (to be with c) by multiplying by term outside of brackets

y = 2(x^{2} – 2.5x + 1.56) + (2)(-1.56) + 1

y = 2(x^{2} – 2.5x + 1.56) – 2.12

– apply perfect square rules to brackets, and write so it resembles vertex form

> square root of first and last term, with symbol of b term

* x^{2} minus, means new bracket sign will be minus

y = 2(x – 1.25)^{2} – 2.12

Quadratics Continued

Vertex Form:

– in factored, standard and vertex, a = same, tells if opens up or down—same parabola, but difference info

– to covert between forms, have to go to standard first

Vertex Form:

y = a(x – h)^{2} + k

where (h,k) is vertex

Transformations:

– any horizontal or vertical shifting of a graph and any stretching/compressing of graph from the parent/baseline graph (including reflecting)

– y = a(x – h)^{2} + k

> x, y = coordinates

> k = vertical movement

> h = horizontal movement

> a = direction of opening, compression, stretch

Stretch:

– narrowing, when a = to more than 1 and less than -1

Compression:

– widening, when a = from 1 to -1

Order:

– order of transformations: horizontal movement, reflection + compression/stretch, vertical movement

Quadratic Formula:

Introduction to Trigonometry

Similar and Congruent Triangles:

Congruent:

– identical in shape, size and angles (same corresponding sides)

– “copy and paste” function

– means congruent to

– SSS – side-side-side, when corresponding sides are equal

– SAS – side-angle-side, if a contained angle and two corresponding sides are equal

– ASA – angle-side-angle, if two angles and the side in between are equal

Similar:

– same shape but different sizes of sides

– corresponding angles are equal but sides are proportional

– “zoom” function

– means similar to

– in ABC and DEF, A = D, and so on

> therefore, AB/DE = BC/EF = AC/DF

– SSS, SAS are same, except proportional

– AA – angle-angle, when two corresponding angles are equal

Scale Ratio:

– scale drawings are a real life example of similar triangles, also how much larger or smaller one triangle is from another

> the constant ratio between corresponding sides is our scale ratio/factor (or “n”)

> n = AB/XY – sides must be corresponding

– if we need to find an unknown side using the scale factor, the length of any unknown side = n x side

> area = n^{2}

> perimeter = n x other perimeter

Connections Between Slopes and Angles:

– slope = rise/run, slope angle is the angle the line makes with the x-axis

– angle of inclination/elevation is when line rises above the horizontal, where angle of declination/depression is when the line rises below

– parallel lines have equal slopes and equal slope angles

– lines with positive slopes have slope angles between 0° and 90°

– lines with negative slopes have slope angles between -90° and 0°

– we find an equivalent angle between 90° and 180° by adding 180°

Primary Trigonometric Ratios:

SOHCAHTOA

– used to find unknown angles/lengths of right triangles

– sin θ = opposite/hypotenuse

– cos θ = adjacent/hypotenuse

– tan θ = opposite/adjacent (aka, the slope angle)

* to find angle, use inverted operations (tan^{-1}, etc.)

Sine Law:

– the ratio of each side, to the sine of the corresponding angle that allows to find any side/angle of a non-right triangle

– in order to sue the law, must be given one side and the corresponding opposite angle and one other angle or side

– each capital letter is an angle and each lowercase is the corresponding opposite side

Cosine Law:

– method used to find the unknown angle or side of a non-right triangle

– must have either all 3 sides given or 2 sides with the contained angle (angle in between)

a^{2} = b^{2} + c^{2} – 2bc cos A

b^{2 }= a^{2} + c^{2} – 2ac cos B

c^{2} = a^{2 }+ b^{2} – 2ab cos C