MPM2D. Key Questions & Concepts. Grade 10Math. peace. love. pi.

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1 MPM2D Key Questions & Concepts Grade 10Math peace. love. pi.

2 Unit I: Linear Systems Important Stuff Equations of Lines Slope à Tells us about what the line actually looks like; represented by m; equation is m = (y2 y1)/(x2 x1) Slope-Intercept Form à Equation of line represented by slope and y- intercept; looks like y = mx + b; m is slope, b is y-intercept Standard Form à Another representation of the equation of a line; looks like ax + by + c = 0; useful for elimination Point-Slope Form à Represents a line using slope and any point on the line; looks like (y y1) = m(x x1); (x1,y1) is a point on the line Solving Equations of Lines Solving Linear Systems à Finding where two line intersect; they can intersect once, never (parallel distinct lines), or infinitely (the SAME LINE) Solving Graphically à Drawing the graphs and seeing where they intersect; put into y = mx + b, plot the y-intercept, and plot the slope Solving by Substitution à Solving algebraically; isolate a variable, substitute it into the other equation, and find the other value; sub back in and find the other point Solving by Elimination à Put the equations into standard form; multiply one or both equations to get the same coefficient for one of the variables, and then add or subtract equations from each other to isolate Other Notes If your slope is the same, check to see if it s the same line! If you get 2=0, the lines are distinct and parallel; 0=0 means THE SAME! peace. love. pi. 1

3 Questions 1. Determine the equation of a line passing through (1,3) and (3, 7), in all three forms of equation. 2. Determine the equation of a line which is parallel to 6x + 3y = 4, and passes through (2, 9). 3. Find the equation of a line passing through (2,8) and perpendicular to 4x 2y = 6 4. Find the point of intersection of y=3x+2 and y=2x +5 graphically. peace. love. pi. 2

4 5. Find the point of intersection of 6x 3y = 9 and 2x + y = 6 using substitution. 6. Find the point of intersection of 2x 5y = 7 and 4x 6y = 9 using elimination. 7. Lindsay seriously needs coffee. She wants 10L of a beverage that contains 42% caffeine. She has a drink with 30% caffeine, and one with 50% caffeine. How many litres of each must be mixed to make the 42% caffeine-infused drinkie? peace. love. pi. 3

5 Unit II: Polynomials and Exponents Important Stuff Polynomials Polynomial à Equation which contains a variable raised to various degrees with one or more terms Term à Part of a polynomial; i.e. 2x; one term is added or subtracted to the next Degree à The highest exponent of a polynomial; if there are brackets; expand the brackets and then find degree Dominant Term à The term which is associated with the degree determining variable (i.e. contains the variable raised to the highest degree) Simplify à Expand out brackets and collect like terms Factoring Factor à Simplifying equations by removing things in common and getting parts of the equation into lower degrees; makes them more manageable Common Factoring à The most common of the factorings; pull out anything in the equation that is a factor Grouping Factoring à Happens when you have four terms in which two sets have things in common; pull out what is in common and group with the terms remaining outside Decomposition Factoring à With trinomials; find two numbers that adds to the middle term and multiplies to the last term by the first; use these numbers to break apart the middle term, and then group Sum-Product Factoring à Trinomial where the first term is 1; in this case, add to the middle and multiply to the last; these two numbers are your terms in the brackets Difference of Squares à A binomial where both terms are a perfect square separated by a negative; take the square root of the first and last term, and separate them with a plus and a minus in different brackets peace. love. pi. 4

6 Other Notes Sometimes you can do more than one type of factoring Always try to common factor first Questions 8. Simplify the following a. 4x(x 5) b. (2x + 3)(5x 1) c. 3x(2x 1)(4x + 3) 9. Determine the degree and state the dominant term of: a. 3x 3 + 2x 8 7x b. 3x 2 (5x 2)(x+6) 10. Factor the following completely and state the method used. a. 3x 3 y 2 + 6x 2 y 12x 4 y 3 peace. love. pi. 5

7 b. 3x 2 6x 45 c. 20x 2 x 12 d. 9x 2 36 e. 16x 2 36x 10 peace. love. pi. 6

8 Unit III: Quadratic Functions Important Stuff Functions Relation à Anything that relates two variables Function à Special type of relation in which no two x-values repeat; if they repeat, it is not a function Vertical Line Test à Way to test if a graph represents a function; if you draw a vertical line and it crosses the graph twice, NOT A FUNCTION Domain à Represents all possible x-values; if it s not part of x, SAY SO! Range à Represents all possible y-values; if it s not part of y, SAY SO! Quadratics Quadratic Function à A polynomial raised to the second degree; forms the shape of a parabola; has a direction of opening up or down and a vertex; depending on the direction of opening, vertex gives a max (if opening down) or min (if opening up), and has an axis of symmetry represented by the x-value (cut it in half and it s symmetrical); also can have, one, two or no x-intercepts Standard Form of Quadratic à f(x) = ax 2 + bx + c; tells you stretch/compression and direction of opening Vertex Form of Quadratic à f(x) = a(x-h) 2 + k; tells you direction of opening, stretch, and (h,k) is your vertex; get to it by Completing the Square Factored Form of Quadratic à f(x) = a(x s)(x t); direction of opening, stretch, and s and t are the intercepts; get to it by factoring Max/Min problem à Word problem where you must create the equation of the quadratic and complete the square to determine the max/min value and/or where it occurs Other Notes For Max/Min problems, you re ALWAYS completing the square If you have a set of points, Domain and Range are just those points peace. love. pi. 7

9 Questions 11. Determine which of the following are functions and state the domain and range. a. b. c. {(3,5), (8,9), (2,4), (4,2), (9,5)} d. {(2,8), (3,8), (4,8), (2,12)} peace. love. pi. 8

10 12. Change the following to vertex form and graph a) y = -3x x - 45 b) y = x 2-10x Find the equation of a parabola with vertex (4,3) and passing through (6,7). peace. love. pi. 9

11 14. Determine the vertex equation of the following without converting to standard form. y=!! (x+3)(x-5) 15. The difference between two numbers is 12. Find the two numbers so that their product is a minimum. peace. love. pi. 10

12 16. A young hooligan is being a nuisance by throwing eggs off of the top of a building 30m tall. If the flight of the eggs is modeled by the equation: h = -0.3t 2 + 3t + 30 Where h is height in meters, and t is time in seconds. Assuming that no citizen or Batman stops the hooligan: a) What is the maximum height that the egg reaches? b) When does it reach this height? c) How long does it take for the egg to reach the ground? peace. love. pi. 11

13 17. Jay-Z and Kanye West, after having great success on their latest studio album, Watch the Throne, decide to take up cattle farming. They have 1200m of fencing in which to construct a rectangular enclosure for their bovine counterparts. Having a fair amount of money, they have managed to get a farm which has a river running through it, and have decided that one side of the enclosure will be bordered by the river. Determine the dimensions that give Kanye and Jay-Z the most area for their non-jungle cows. peace. love. pi. 12

14 18. Vanessa, after starting a most successful dollar fruit business, decides to buy the Phoenix Coyotes, who had no fans in their previous city, and move them to Toronto, creating a second NHL team in Toronto to compete with the completely incompetent Leafs. Vanessa has a stadium which seats 25,000 fans, and finds that she sells out when she charges $30/seat. She also finds that if she increases the price by $5, she loses 250 fans. a) What price should Vanessa charge for admission to a game in order to maximize her revenue? b) How much revenue does Vanessa make at this price? peace. love. pi. 13

15 Unit IV: Solving Quadratic Equations Important Stuff Solving Quadratics Solving à Means finding the x-intercepts of a parabola; can be one, two or no solutions; we also call if finding the zeroes Solve by Factoring à Factor the quadratic, set each bracket equal to zero; these are the solutions Solve by Graphing à Graph the parabola; state where it hits the x-axis Solve by Quadratic Formula à Used for quadratics that are unfactorable; the formula is x =!!±!!!!!" ; plug your values in from standard form to get!! your zeroes The Discriminant à The part of the Quadratic Formula under the square root sign (b 2 4ac); if it s positive, yields two solutions; 0 yields 1 solution; negative yields one solutions Other Notes It s all factoring all over again Don t forget to try common factoring first! For word problems, make two equations and associate them together Questions 19. Solve the following quadratic equations by factoring: a. y = 5x 2 45 b. y = x 2 + 8x + 16 peace. love. pi. 14

16 20. Solve the following equation graphically: y = -3(x + 4) Solve the following using any method a. y = 6x 2 + 5x 6 b. y = -3x x + 7 peace. love. pi. 15

17 22. Bored after successfully completing his bike jump, Joel decided to build a model rocket, and launch it in an attempt to get across a 90m wide canyon so that like the chicken, he could get to the other side. The flight of the rocket is modeled by the equation: h = -0.1d 2 + 6d + 1 Where h is height in metres and d is distance travelled, also in metres, will Joel s rocket make it across the canyon? 23. Vanessa, after constructing a new garden in which to grow fruits for her dollar fruit stand, decides that there should be a deck around her garden where she can have a café which serves tea-na coladas, without alcohol of course. If the garden has dimensions of 10mx20m and she wants the deck to have twice the area of the garden, how wide should she make her deck? peace. love. pi. 16

18 Unit V: Trigonometry Important Stuff Basic Triangles Similar Triangles à Triangles with same angles; the sides are all ratios of each other, allowing you to determine lengths of the other sides Pythagorean Theorem à For a right angled triangle; has the formula a 2 + b 2 = c 2 ; a and b are the legs, c is the hypotenuse across from the hypotenuse Trigonometric Ratios SOH CAH TOA à The primary trig ratios; stands for SINE (Sin x = opp/hyp), COSINE (Cos x = adj/hyp), and TANGENT (Tan x = opp/adj); label the triangles in relation to the angle, like so: Opp Hyp x Adj Non-Right Triangles Sine Law à Used when there are two sides and two opposite angles; has formula!"#!! =!"#!! =!"#!! Cosine Law à Used when there are three sides and one opposite angle; looks like a 2 = b 2 + c 2 2(b)(c) Cos A Label triangles like so: B a C c b A Other Notes Remember, the trig ratios ONLY WORK ON RIGHT ANGLE TRIANGLES! peace. love. pi. 17

19 Questions 24. Solve x and y. 3 4 x y Find x. a. x 42 o 5 b. 10 x 8 peace. love. pi. 18

20 c. 6 x Determine the value of x x 27. Solve for x o x peace. love. pi. 19

21 28. Find x. 8 x 5 10 peace. love. pi. 20

22 Unit VI: Analytic Geometry Important Info Line Properties Parallel Lines à Lines that have the same slope Perpendicular Lines à Lines where the slopes are negative reciprocals; flip the slope and change the sign Intercepts à Where it hits the axis Shortest Distance between Point and Line à Found by determining the intersection of line and perpendicular line containing point, and finding length Geometry Tools Line Segment à A piece of a line defined by two coordinates Midpoint à The middle of a line segment, found by formula M = (!!!!!,!!!!! )!! Length Formula à Finds the length of a line segment (or distance); has formula l = x! x!! + y! y!! Triangle Tools Altitude à A line which is perpendicular to one side of a triangle and passes through the opposite point; uses perpendicular slope and a vertex Median à A line which passes from the midpoint of one side to the vertex of the of the opposite angle; uses midpoint and a vertex Perpendicular Bisector à Line which goes through the midpoint of a side and is perpendicular to it; uses midpoint and perpendicular bisector Other Notes All of these concepts require slope; start by figuring out how to find your slope peace. love. pi. 21

23 Questions 29. Determine the equation of a line perpendicular to 2x 4y = 8, and passing through (2,4). 30. Find the equation of a line parallel to y = 3x 2 and passing through (4,7). 31. Determine the equation of a circle centred at the origin and with a radius of 6. peace. love. pi. 22

24 32. Find the shortest distance between the point (5,8) and the line created by connecting (8,6) and (4,-2). (Hint, you ll need to use the perpendicular slope to find the shortest distance) 33. A triangle consists of points a(4,8), b(5,12) and c(8,6). Determine the equations of the altitude, perpendicular bisector, and median on line ac. peace. love. pi. 23