11 College Math Final Exam Review
Topics • Part 1: Trigonometry • Part 2: Quadratic Relations • Part 3: Probability • Part 4: Statistics • Part 5: Exponents • Part 6: Personal Finance • Part 7: Measurement
Agenda This week … Monday: Right-angle Trigonometry (SohCahToa) Tuesday: Sine Law & Cosine Law Wednesday: Quadratics (Algebra & The Parabola) Thursday: Quadratics (3 Forms & Applications) Friday: Probability & Statistics Next week … Monday: Exponents Tuesday: Personal Finance Wednesday: Geometry
Part 1: Trigonometry Monday: Right-angle Trigonometry • Labelling: Opposite, Adjacent, Hypotenuse • SOHCAHTOA: Opposite, Adjacent, Hypotenuse • Right-angle Applications Non-Right Angle Trigonometry Tuesday: • The Sine Law • The Cosine Law • Applications
Right-Angle Trigonometry Primary Trigonometric Ratios: Sine, Cosine & Tangent Example 1: Evaluate to 3 decimal places. sin 650 = cos 1240 = tan 3410 = -0.344 0.906 -0.559 ANGLES Example 2: Solve for the angle to the nearest degree. sin R = 0.25 cos B = 0.92 tan Q = 1.54 Recall: When solving for an angle we must use the inverse functions! B = cos-10.92 Q = tan-11.54 R = sin-10.25 = 14o = 23o = 57o
When do we use these? When we want to solve for an angle or a side in a Right-angle Triangle. What is the first thing that we must do when solving a right angle triangle? LABLE THE SIDES: OPPOSITE, ADJACENT, HYPOTENUSE Recall: Labelling depends upon the reference angle! HYPOTENUSE HYPOTENUSE ADJACENT OPPOSITE OPPOSITE ADJACENT
Recall: SOHCAHTOA Hence, the primary trig ratios for angle A below are: A sinA = HYPOTENUSE ADJACENT cosA = tanA = B C OPPOSITE
Example 1: Determine the length of side b to one decimal place. SOHCAHTOA COSINE H 11.5 m O 42o A Variable on top b → MULTIPLY
Example 2: Determine the length of side b to one decimal place. SOHCAHTOA 37o TANGENT H A b Variable on bottom 7 cm O → DIVIDE
Example 3: Determine the length of angle A to the nearest degree. SOHCAHTOA A SINE H 2.5 in A B C Solving for an angle 1.2 in O → INVERSE 2nd sin (1.22.5)
Joey is standing 150 ft from the base of a building. The angle of elevation to the top of the building is 29o. How tall is the building. Example 4: SOHCAHTOA Right Angle Triangle Diagram: H BUILDING Variable on top O x → MULTIPLY 29o 150 ft A JOEY the building is 83 feet tall.
By the end of today … I can label the sides a right angle triangle OPPOSITE, ADJACET AND HYPOTENUSE in reference to a certain angle. I can use SOHCAHTOA to select the appropriate Primary Trig Ratio based on the given information. I can solve for the unknown side or angle in a right-angle triangle using SOHCAHTOA. I can create a diagram from the words of a typical right-angle trig application question and apply SOHCAHTOA to solve the problem.
Non Right-Angle Trigonometry Solving triangle that are NOT right angles NO Can we use the Pythagorean Theorem? NO Can we use SOHCAHTOA? Then ... what can we use? SINE LAW COSINE LAW
Before we begin let's recall a few things about triangles: #1 Sum of the angle in a triangle: B A + B + C = 180o C A #2 Labelling Conventions: Q · Angles are labelled with UPPER CASE letters. p · Sides are labelled with LOWER CASE letters. r · OPPOSITE sides and angles correspond. R P q
The Sine Law Solving for a side: B a c Solving for an angle: C A b Recall: Look for the OPPOSITE SIDE-ANGLE PAIR! i.e. Are you given A & a, B & b or C & c? If you are given two angles, always start by finding the third angle (i.e. 1800 minus the other two)
Example 1: Determine the length of side b to one decimal place. B 83o Only use 2 fractions … 12 cm c 65o 32o C A b Given: A & a → THE SINE LAW REFLECT 12sin83o = sin65o The size of opposite side-angle pairs should correspond (i.e. the biggest side should be across from the biggest angle).
Example 2: Determine the measure of angle A to the nearest degree. B Note: angles are on top! 3.1 in 2.3 in 47o C A b Given: C & c → THE SINE LAW 3.1sin47o = 2.3 2nd sin ANS
The Cosine Law Solving for a side: B a Solving for a angle: c C A b Recall: look for a side-angle-side “sandwich” Solving for a side → look for all 3 sides Solving for an angle → Use cosine law if you can’t use sine law (i.e. no Side-Angle pair)
Example 3: Determine the length of side b to one decimal place. B 75o Change letters to solve for b… 11 m 12 m C A b Side-angle-side sandwich = ANS → THE COSINE LAW
Example 4: Determine the measure of angle A to the nearest degree. B Always subtract the side across from the angle you are solving for. 19 m 26 m C A 15 m Given: All 3 sides → THE COSINE LAW 2nd cos (-90 570)
Two planes took off from Pearson International Airport at the same time. The first plane is travelling due west at a speed of 168 Km/hr. The second plane is travelling due east at a speed of 156 Km/hr. How far apart are the planes after 2 hours? Assume the angle between them is 125o. Example 5: Diagram: Non Right-Angle Triangle Side-Angle-Side Sandwich Cosine Law Plane 1 Plane 2 336 Km 312 Km (168 x 2) (156 x 2) AIRPORT the planes are 574.9 Km apart after 2 hours.
By the end of today … I can identify when to use the Sine Law and apply it to solve for the unknown side or angle in a triangle. I can identify when to use the Cosine Law and apply it to solve for the unknown side or angle in a triangle. I can create a diagram from the words of a typical non right-angle trig application question and apply the Sine Law or the Cosine Law to solve the problem.
Part 2: Quadratics Wednesday: Algebra & The Parabola • Expanding from Factored Form to Standard Form • Expanding from Vertex Form to Standard Form • Factoring: Common Factoring and Simple Trinomials • Key features of the Parabola • Vertex form and transformations from Thursday: Graphing, Three Forms & Applications • Graphing from Vertex Form • Identifying the key features of a parabola from the three forms. • Applications (projectile & revenue)
Algebra - Expanding Convert the following equations form Factored Form to Standard Form. b) a) Example 1: Example 2: Convert the following equations form Vertex Form to Standard Form. b) a) +5
Algebra - Factoring Example: Convert the following equations form Standard Form to Factored Form. a) x b) x
The Parabola Axis of Symmetry -intercept -intercept -intercept Vetex (Max/Min)
Transformations from Vertex Form: • horizontal translation v • vertical translation • vertical stretch/compression by a factor of Step Pattern: 1, , • reflection in the -axis.
Transformations from State the parabola represented by has been transformed from the graph of . Example 1: Example 2: • Translated 3 units right. • Translated 7 units up. • Vertical stretch by a factor of 2. • Reflected in the x-axis. Determine the equation of the parabola thathas been transformed from the graph of as follows: • Translated 3 units left. • Translated 7 units down. • Vertical stretch by a factor of 5.
By the end of today … I can convert a Quadratic Relation from Factored Form to Standard Form by Expanding and collecting like terms. I can convert a Quadratic Relation from Vertex Form to Standard Form by Expanding and collecting like terms. I can convert a Quadratic Relation from Standard Form to Factored Form by common factoring and/or simple trinomial factoring. I know and can identify the key features of a parabola. I can identify the transformations of a parabola form the graph of when given the equation in vertex form. I can write the equation of a Quadratic Relation in Vertex Form when given the transformations from the graph of .
Graphing from Vertex Form Recall from yesterday … • horizontal translation v • vertical translation • vertical stretch/compression by a factor of Step Pattern: 1, , • reflection in the -axis.
Graphing from Vertex Form Graph the equation, on the graph below. Example: Vertex: Step Pattern:
Recall From yesterday …. Axis of Symmetry -intercept -intercept -intercept Vetex (Max/Min)
Three Forms All forms: -- d + + step pattern: 1, , Vertex Form: vertex: Standard Form: -intercept: Factored Form: -intercept(s):
Three Forms Complete the chart of the key features of the following quadratic relation. Example: DOWN -3, -9, -15 -36 -3, -7 (-5, 12)
Application - Projectile A football is kicked upwards. Its height is described by the equation where is the height measured in metres and is the times measured in seconds. a) What is the height of the ball after 3 seconds? the ball is 20 metres high after 3 seconds. b) When does the ball hit the ground? -intercept(s): the ball hits the ground after 8 seconds.
Application - Projectile A football is kicked upwards. Its height is described by the equation where is the height measured in metres and is the times measured in seconds. c) When does the ball reach its maximum height? -intercept(s): Middle: the ball reaches its maximum height at 5 seconds. d) What is the balls maximum height? the ball reaches a maximum height of 36 metres.
Application - Projectile A football is kicked upwards. Its height is described by the equation where is the height measured in metres and is the times measured in seconds. e) Sketch a graph of the projectile of the football.
Application - Revenue The owner of a waste management company is given a graph that shows the relation between profit and mass of garbage his company recycles. The graph is shown below:
Application - Revenue The owner of a waste management company is given a graph that shows the relation between profit and mass of garbage his company recycles. The graph is shown below: • a) What is the maximum profit possible? $180,000 • b) How many tonnesof garbage must be recycled to produce the maximum profit possible? 500 tonnes • c) A company is said to “break even” when revenue equals expenses, or when profit equals zero. How many tonnes of garbage must be recycled to break even? 200 tonnes OR 800 tonnes
By the end of today … I can graph a Quadratic Relation from Vertex Form. I can identify the y-intercept and step-pattern of a parabola given the equation in Standard Form. I can identify the x-interceptsand step-pattern of a parabola given the equation in Factored Form. I can identify the vertexand step-pattern of a parabola given the equation in Vertex Form. I can create a diagram from the words of a typical application question (i.e. projectile, revenue) and select the appropriate information form the equation to solve the problem.
Probability Experimental Probability Theoretical Probability VS • The likelihood (odds) of an event occuring. • The number of times an event actually happens out of a certain number of trials. Whereas…. For example …. • The Theoretical Probability of rolling a 5 on a dice is ALWAYS … • The Experimental Probability of rolling a 5 on a dice is …
Theoretical Probability Example 1: A card is drawn from a standard deck of cards. What is the probability that it is a 9 of any suit or a black face card? Leave your answer as a fraction in lowest terms. A jar contains 5 red balls, 8 green balls and 2 black balls. If you select on ball from the jar, what is the probability of NOT getting a green ball? Leave your answer as a fraction in lowest terms. 4 Probability: # of 9’s: Lowest Terms ← 6 # of black face cards: 10 a 52 Total # of cards: the probability of drawing a 9 or a black face card is . Example 2: Probability: 8 # of green balls: 15 Total # of balls: ← Already in Lowest Terms the probability of NOT getting a 9 green ball is .
Statistics Measures of Central Tendency: Measures of Spread: Range: Biggest - Smallest The “average”. Mean: Standard Deviation: • Add them all up, press equal and then divide by how many there are. • The typical distance a particular value is from the mean. The middle. Median: Quartiles: • Line up the data and find the middle # or the average of the 2 middle #’s. • 3 #’s that divide the data into 4 equal parts. • The 2nd quartile is the median. The most frequent. Mode: Interquartile Range: • Note: there can be more than one mode. • 3rd Quartile – 1st Quartile
Example: Mackenzie recorded the following daily temperatures for 8 days in June: 23 25 17 16 27 25 32 12 a) Calculate the Mean, Median, Mode and Range: Mean: Median: Middle Mode: Range:
Example: Mackenzie recorded the following daily temperatures for 8 days in June: 23 25 17 16 27 25 32 12 b) Find the three quartiles and state what they are: 1st Quartile: Middle of 1st half 2nd Quartile: Middle → Note: this is the same as the Median. 3rd Quartile: Middle of 2nd half c) Determine the Interquartile Range:
Standard Deviation An engine part is manufactured with a mean weight of 900g and a standard deviation of 9g. Parts are rejected if they are not within one standard deviation of the mean. How many of the parts with the following weights would be rejected? 890g 899g907g905g911g895g905g x x +9g -9g 900g 909g 891g 3 of the parts would be rejected: 890g, 911g & 895g.
By the end of today … I understand and can recognize the difference between experimental and theoretical probability. I can use the respective formula to calculate the experimental or theoretical probability of various situations (i.e. rolling a die, drawing cards from a deck, etc.) I can calculate the mean, median, mode and range of a given set of data. I can identify the THREE quartiles in a given set of data.
Exponents Exponent Laws: 1. Multiplication 2. Division Don’t forget … 3. Power of a power 4. Negative exponents
Exponents Examples: Simplify by expressing as a single power. Write all answers with positive exponents where required. c) 1 b) a) e) d)
Exponential Curve Decreasing for … Increasing for … →The larger the value of a the steeper the slope. →The smaller the value of a the steeper the slope. Always crosses the -axis at 0.