MPM 2D Course Review

1. MPM 2D Course Review Unit 1: Linear Systems

2. I can graph a line • 3 ways: • Table of values • x- and y-intercepts • y-intercept and slope • Ex. 1) y=-3x+2

3. I can graph a line • 3 ways: • Table of values • x- and y-intercepts • y-intercept and slope • Ex. 2) 4x-3y=12

4. I can graph a line • 3 ways: • Table of values • x- and y-intercepts • y-intercept and slope • Ex. 3) y=(2/3)x-5

5. I can translate a word problem into algebra • What important, defining information are you missing? THIS WILL HELP YOU DEFINE YOUR VARIABLES • Hint: The question at the end will direct you to at least one of the variables • EXAMPLE: Karl owns a small airplane. He pays $50/h for flying time and $300/month for hangar fees at the local airport. If Karl rented the same type of airplane at a flying club, it would cost him $100/h. When will the monthly cost of owning and renting be the same?

6. I can solve a linear system by graphing • Karl’s situation: • y1=50x+300 • y2=100x

7. I can determine if a point is the solution to the system • …using substitution! • We found the POI for Karl was (6, 600) on the graph. • How do we test our answer? • Plug it into both equations! • y1=50x+300 • y2=100x

8. I can determine the number of solutions by looking at the equations in the system • 1 solution • Different slope • No solutions • Same slope, • different y-intercepts • Infinite solutions • Same slope, • same y-intercept

9. I can solve a linear system by substitution • and • CHECK YOUR ANSWER!

10. I can solve a linear system by elimination • and • CHECK YOUR ANSWER!

11. I can solve problems using linear systems • A weekend at a lodge costs $360 and includes 2 nights’ accommodation and 2 meals a day. A week costs $1200 and includes 7 nights’ accommodation and 10 meals. What is the cost of one night and one meal? How much would it cost for five nights and 4 meals? • CHECK YOUR ANSWER!

12. MPM 2D Course Review Unit 2: Analytic Geometry

13. I can calculate the length of a line segment, given its endpoints • Distance formula: • Find the distance between (-7, 1) and (5, -2)

14. I can find the midpoint of a line segment • Midpoint formula: • Ex 1. Find the midpoint between (-4, 6) and (8, -2)

15. I can find the midpoint of a line segment • Midpoint formula: • Ex 2. Find the other endpoint of a line segment if one endpoint is (-4, -2) and the midpoint of the line segment is (2, 6).

16. I can solve problems involving midpoints, medians, and perpendicular bisectors • Median: a line that joins a vertex of a triangle to the midpoint of the opposite side. • Perpendicular Bisector: of a line segment is the line that is perpendicular to the line segment and passes through the midpoint of the line segment.

17. I can solve problems involving midpoints, medians, and perpendicular bisectors CONT’D • STEPS TO FIND EQN OF MEDIAN: • Calculate the midpoint of the line opposite the vertex of interest • Calculate the slope of the line connecting the vertex of interest to that midpoint (ie. The median) • Sub the slope into the eqn for a line: y=mx+b • Plug in (x, y) (either the midpoint or the vertex of interest) • Solve for b • State equation with slope (m) and y-intercept (b) • Find the equation of the median line from vertex C in triangle ABC if the coordinates of the vertices are A(-3, 3), B(2, -5), and C(5, 2)

18. I can solve problems involving midpoints, medians, and perpendicular bisectors CONT’D • STEPS TO FIND EQN OF PERPENDICULAR BISECTOR: • Calculate the midpoint of the line segment • Calculate the slope of the line segment • The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment it is bisecting • Plug the slope and the midpoint into y=mx+b • Solve for b • State equation with slope (m) and y-intercept (b) • Find the equation of the perpendicular bisector of the line segment from A(1, 1) to B(5, 3)

19. I can classify triangles given the coordinates of the vertices • The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Determine what kind of triangle it is.

20. I can classify triangles given the coordinates of the vertices • The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Determine what kind of triangle it is.

21. I can verify properties of geometric figures algebraically • The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Show that the median from vertex C is half as long as the hypotenuse.

22. I can verify properties of geometric figures algebraically CONT’D • Classify the shape with vertices at A(-4, 2), B(2, -5), C(7, -3) and D(1, 5)

23. I can verify properties of geometric figures algebraically CONT’D • PQRS is a rhombus with vertices at P(3, 3), Q(0, 1), R(3, -1), and S(6, 1). • Verify that its diagonals bisect each other at right angles.

24. I can determine the shortest distance from a point to a line • …the shortest distance from a point to a line is always the PERPENDICULAR PATH from the point to the line.

25. I can determine the equation of a circle centered at the origin, given the circle’s radius • Equation of a circle: • State the equation of the circle: • (a) centre origin, radius of 8 • (b) centre origin, radius of

26. I can determine the radius of a circle centered at the origin, given the circle’s equation • Length of a circle’s radius: • A circle has the equation . Where is its centre? What is its radius?

27. I can sketch a circle, given its equation • A circle has the equation . Sketch its graph.

28. I can determine the equation of a circle, given a point through which it passes • Find the equation of the circle passing through (-5, 12)

29. STUDY • Shapes and their names and properties • Important terms (midpoint, median, perpendicular bisector) • Formulas

30. MPM 2D Course Review Unit 3: Quadratics (Standard & Factored Form)

31. I can simplify expressions involving exponents

32. I can expand & simplify polynomials • Distributive property •  • FOIL (first, outside, inside, last) •  Ex.

33. I can factor using GCF

34. I can factor differences of squares

35. I can factor

36. I can factor • 20

37. I can solve a quadratic equation by factoring

38. I can identify the key features of a graph of a parabola

39. I can graph a parabola (determine the zeros and vertex ) from

40. I can find the equation of a quadratic given its zeros and another point on the parabola • A parabola has zeros at (-2, 0) and (4, 0), and passes through (2, 16). Find its equation.

41. I can solve problems involving quadratics • When Kermit the Frog makes a giant leap from one lily pad to another, he follows a parabolic path. Kermit is in the air for 6 seconds before he makes a safe landing. Kermit knows that after 2 seconds, he is 72 cm high. How high is Kermit at his greatest height?

42. I can solve problems involving quadratics • A design engineer uses the equation to model an archway for the entrance to a fair, where h is the height in metres above the ground, and d is the horizontal distance from the centre of the arch. • How wide and tall is the arch? • For what values of d is the relation valid? Why? • If a width of 2.5 m is needed per line-up at the entrance, how many line-ups can there be?