Math 10C Year End Review Linear Relations and Functions PDF

Name: ____________________ Math 10C Year End Review Linear Relations and Functions Coordinate Plane Coordinate Points are always listed as (x,y) Continuous Data: Values between data points make sense - connect the dots Eg: Grow...

Name: ____________________ Math 10C Year End Review Linear Relations and Functions Coordinate Plane Coordinate Points are always listed as (x,y) Continuous Data: Values between data points make sense - connect the dots Eg: Growth of a student over time Discrete Data: values between data points do not make sense – do not connect the dots Eg: Revenue at a concert when tickets are sold for $20 each Independent Variable = Input = Domain Dependent Variable = Output = Range There are multiple ways to represent data: 1. Graphs 4. Words Eg: the cost of driving a car is related to the speed at which it is driven. 5. Mapping 2. Table of Values 6. Equation 3. Ordered Pairs 7. Function Notation x-intercept: the point where the graph crosses the x-intercept (when y=0) y-intercept: the point where the graph crosses the y-intercept (when x=0) Example: calculate the x and y intercept of 𝑦 = 2𝑥 + 4 x-intercept y-intercept Domain: the complete set of possible values of the independent variable (x). Range: the complete set of all possible values of the dependent variable (y). Example: determine the domain and range of the following graphs. Functions: a special type of relation where the inputs have exactly one output. Example: determine if the following relations are functions. Function Notation: special notation that is used for functions 𝑓(𝑥) 𝑜𝑟 𝑔(𝑥) 𝑜𝑟 ℎ(𝑥) … are the “names” of the functions. Example: Use the function 𝑓(𝑥) = 2𝑥 + 1 to determine the follow: a. 𝑓(2) b. 𝑓(𝑥) = 7  𝑟𝑖𝑠𝑒 Slope: the “steepness” of a graph 𝑚= 𝑟𝑢𝑛 𝑦2−𝑦1 If given two coordinate points, use 𝑚 = 𝑥2−𝑥1 Example: calculate the slopes of the following graphs Collinear: points that have the same slope Parallel Lines: have the same slope, different y-intercepts Perpendicular Lines: two lines have slopes that are negative reciprocals of each other. Example: Given the two points P(4,8) and Q(6, -3), determine the slope of another line segment that is a. Parallel to PQ b. Perpendicular to PQ Rate of Change: a way to describe the slope in word problems. Example: If it costs $25 to rent a car and 10 cents per kilometer, create a graph to show the scenario. a. Calculate the rate of change. What does it represent? b. Determine the y-intercept. What does it represent?  Extra Relations and Functions Practice 1. A basketball team is hosting a banquet for its players and friends. The caterer charges a set-up fee of $200 plus $10 for each person. The equation C = 10 p + 200 represents the cost C of the banquet for n people attending. a. Sketch the graph of the function on the given grid. b. Determine the cost for 30 people attending. c. If the total cost is to be no more than $700, how many people can attend? d. Determine the slope of the graph and explain its significance. e. What is the y-intercept, and what is its significance? f. What is the domain and range of the function? Are there any restrictions on either? Explain. 2. Given that f ( x) = 2 x 2 + 3x − 6 , determine the following: 3 a. f(2) b. f   c. f ( 5) 5 3. Given the function f(x) = 3x +6, find x for the following questions. a. f(x) = 21 b. f(x) = -9 c. f(x) = 3 4. The graph below shows the cost of movie tickets at a local cinema. a. Why are the points on the graph not joined? b. Describe in words how the variables are related. c. Write an equation to show the relationship between cost and number of tickets. 5. Find the domain and range of the following relations. Each space on the grids is 1 unit. A. B. C. 6. The cost of publishing books includes an initial cost of $2000 and a cost of $20 per book after. Books can only be ordered in sets of 100. a. Make a table of values for up to 1000 books. b. Graph the relation on the given grid. # of books Cost ($) 0 2000 1000 c. Find an equation that represents the relation. d. Should the points on the graph be connected? Explain. 7. Consider each line segment shown below. Indicate whether the slope is positive, negative, zero, or undefined. y2 − y1 8. Use the slope formula, m = to find the slopes of the line segments with endpoints given x2 − x1 below: a. A ( 2,1) , B ( 5, 3 ) b. C ( −3, 4 ) , D ( −1, −2 ) c. G ( 4, −2, ) , H ( 5, 4 ) d. M (1, −2 ) , N (1, 3 ) 9. Determine the value of x or y if the line segment joining the given points has the slope given below. 4 2 a. ( 6, y ) , ( 9,10 ) , m = − b. ( 3, 5 ) , ( x, 3) , m = 3 5 c. ( 6,19 ) , ( x,5 ) , m = 7 d. ( 6, y ) , (1, 3) , m = 0 10. The slopes of several line segments are given below. 1 1 1 mAB = 2, mJK = , mMN = −2, mXY = − , mGH = , mCD = −0.5 2 2 2 a. Which line segments are parallel? b. Which line segments are perpendicular? 11. Determine the slope of each line segment with endpoints given below. Identify which line segments are parallel and which are perpendicular. a. A ( 5,3 ) , B ( 0, 0 ) b. C ( 6, −1) , D (1, 2 ) c. E ( −5, −8 ) , F ( −2, −6 ) d. G ( −4, 3 ) , H ( −1, −2 ) 12. Determine the value of x so that the line segments with endpoints R ( x, 5 ) and S ( 6, 3 ) is parallel to the line segment with endpoints T ( −1, −6 ) and U ( 3, 2 ).  SOLUTIONS 1. b. 500 c. 50 d. 10 – price increases by $10 per person e. $200 – price charged by caterer initially f. D{0