Lecture Note
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Virtual High SchoolCourse
MPM2D | Principles of MathematicsPages
12
Academic year
2023
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MPM2D – Grade 10 Academic Math Exam Notes Modeling With Linear Equations: – linear is a straight line, have an x/y intercept –The point of intersection is where two or more lines intersect to form a linear system, andthere is a solution there. – point of intersection is a single point that satisfies both equations – breakeven problems: > revenue = profit – cost, but when revenue = 0, that is break even 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 bothlines > x1 = x2 and y1 = y2 – steps: > number equations > isolate x or y on either side (y is easier) > set one equation as equal to the other (y1 = y2), 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 findeliminated 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 variableare 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 aspossible > remaining points are equal on either side Coordinate Geometry Finding the Length of the Line: d = √(x 2 – x1)2 + (y2 – y1)2
Equation of a Circle: x2 + y2 = r2 – r is radius or distance from center of circle to edge or half diameter – works when center is (0,0) Midpoint: M = x1+x2, y1+y2 2 2 – median: a line segment that goes from the vertex of one point to the opposite side andcrosses 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 vertexto get equation Quadratic Relations Quadratics: – in quadratic relations, the 2nd 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 – 2nd difference is difference in the 1st 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 pointswith same y coordinate – 2nd 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 isa 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 otherand by two Factoring: – ask: common factor? 1st and 2nd 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 – x2 + 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. 2x2 + 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 eachother – take square root of both then write as (x + s)(x – s) – eg. 20a2 – 180 > 20(a2 – 9) > 20(a + 3)(a – 3) Perfect Squares: – a2x2 +/- 2abx + b2 – 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 factoredcompletely – 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 = 2x2 + 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 = 2x2 – 5x +1 – factor out the coefficient of the x-squared term, and leave c alone y = 2(x2 – 2.5x) + 1 – take half of factored b term, then square it—then add and subtract to keep equation thesame y = 2(x2 – 2.5x + 1.252 – 1.252) + 1 y = 2(x2 – 2.5x + 1.56 – 1.56) + 1 – take out subtracted term (to be with c) by multiplying by term outside of brackets y = 2(x2 – 2.5x + 1.56) + (2)(-1.56) + 1 y = 2(x2 – 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 * x2 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, butdifference info – to convert 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 fromthe 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, verticalmovement 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 smallerone 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 = n2 > 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 ofdeclination/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 inverse operations (tan-1, etc.) Sine Law: – the ratio of each side, to the sine of the corresponding angle that allows to find anyside/angle of a non-right triangle – in order to sue the law, must be given one side and the corresponding opposite angle andone 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) a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C
MPM2D – Academic Math Exam Notes
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