Math Secondary 4 CST Topic 4. Functions

Transcription

1 Quadratic Function Functions The graph of the basic quadratic function is drawn in the Cartesian plane below. We call the curve a. The function will pass through (, ) y = x 2 x y The rule of the quadratic function is: Where a is responsible for If a > 0 the graph is If a < 0 the graph is

2 Finding the Rule To find the rule of a quadratic function we need exactly coordinate Steps: 1) Plug in that is NOT (, ) 2) solve for 3) Rewrite equation with a Example 1: Find the rule of the quadratic function Example 2: Find the rule of the quadratic function passing through (-1,3) Example 3: The length of a rectangle is twice as long as its width. What is the quadratic function that relates the area to its width? Example 4: The path of a skateboard as it rolls down a road can be represented by a quadratic function, where x represents the time in seconds and y represents the distance in meters travelled by the skateboard. If the skateboard rolled, 2 meters after 4 seconds find the rule that represents this situation.

3 Solving for x and y To solve for y we simply plug in x and simplify Example 1: Using the rule found in example 4, find the distance the skateboard would have traveled after 5 seconds. Example 2: (4,?) To solve for x we: 1) Plug in y 2) Divide by a 3) both sides of the equation 4) Don t forget! Final answer has two options. Example 1: Using the same graph as above solve for x when y = 9 Example 2: Using the same example as before where the quadratic function y = 0.125x 2 represents the path of a skateboard as it rolls down a road. Find the number of seconds elapsed after the skateboard is at a distance of 28 meters.

4 Word Problems Example 1: The vertical distance travelled by a free-falling object as a function of time elapsed is a quadratic function. Given the table of values below, what is the vertical distance travelled by an object after 5 seconds? Time (s) Distance (m) Example 2: Higgins is looking for a new designer rain barrel to catch water from his downspouts. There are two stores that carry these rain barrels, Lee Vallet Tools L(x) and Home Depot H(x). Both function are in the form f(x) = ax 2 where: x represents the radius of the barrel in decimeters L(x) represents the cost of the barrel in dollars at Lee Valley Tools H(x) represents the cost of the barrel in dollars at Home Depot Higgins has noticed the following pricing tables in the online catalogues of each store. Radius (dm) L(x) $ Radius (dm) L(x) $ The model Higgin s is looking for has a radius of 20 dm. Where should he purchase his barrel if he wants to spend the least amount of money?

5 Exponential Function The exponential function is used to represent exponential growth in, or. The rule of the exponential function is Parameter a is the Parameter c is responsible for the If c > 1 the function If 0 < c < 1 the function Therefore there are two possibilities for the graph: Solving for y a) y = 5(2) x x y b) y = 0.5(3) x x y

6 Finding the rule Similar to the quadratic function we only need point and the initial value to find the rule. 1) First we must determine a from the graph 2) Second we plug in our point to solve for c Example 1: Find the rule of the exponential function to the right Example 2: Find the rule of the exponential function to the right Example 3: Find the rule of the exponential function that corresponds to the table of values below.

7 Finding the rule given Word Problems The rule we can use when solving a word problem is: Where a is the Where c is what occurring i.e.: doubling, tripping etc. * if there is a % in the problem then c = 1 ± % if % is it is 1+ if % is it is 1 - Where b is the Example 1: Scientists are studying the growth of bacteria. The bacteria double every hour. Initially, there are 1000 bacteria. Find the rule that represents this situation.

8 Example 2: An ant colony initially contains 4 ants. Researchers note that the number of ants in the colony triples every 2 weeks. Find the rule that represents this situation. Example 3: A strain of bacteria triples 3 times a day. You start with a population of 150 bacteria. Find the rule that represents this situation. Example 4: A strain of bacteria doubles every 30 minutes. You start with a population of 50 bacteria. Find the rule that represents this situation. Example 5: Your parents win the lottery. They put $ dollars in an investment plan that earns an annual interest rate of 3.5%. Find the rule that represents this situation. Example 6: It is well know that cars depreciate once they leave the lot. A brand new Honda Civic costs $20,000. It depreciates at a rate of 4.5% per year. Find the rule that represents this situation. Solving for x and y ***Using the example on the previous page Example 1: Rule y = a) How many bacteria are present after 2 hours b) How many hours pass until there are bacteria

9 Example 2: Rule y = a) How many ants will you have after 10 weeks? b) How many weeks pass if you have 108 ants? Example 3: Rule y = a) How many will you have after 6 days? b) How many days pass until there are bacteria? Example 4: Rule y = Determine the number of bacteria after 3 hours. Example 5: Rule y = Determine how much you earn in interest after 5 years. Example 6: Rule y = In what year will it be worth $ ?

10 Step (Greatest Integer) Function The greatest integer function, also known as the function because of the shape of its graph. We will not be learning the rule of the step function, you will only be required to know how to the graph. The closed dot The open dot means means They will always appear above each other because a function cannot be equal to the same number at the same time. Therefore, we should only pay attention to the dot. The line means that the value remain constant for an extended period. You will be asked to solve: Example 1: For f(4) =? This means what is when. Example 2: For f(11) Example 3: For f(x) = 9 This means what is when. Example 4: For f(x) = 17

11 Word Problems: Example 1: A parking garage is free for the first hour. After that you pay $4 for every 2 hours you remain parked. a) Represent the situation on the adjacent cartesian plane b) How much will you pay after 6 hours? c) How many hours did you park if you pay $16? Example 2: At a discount store, you get a $2 coupon for every $5 you spend before taxes. Graph the amount you get in coupons as a function of the value of you purchase. a) Represent the situation on the adjacent Cartesian plane b) If you spend $30 what value of coupons will you get? c) You received a coupon with a value of $6, how much did you spend?

12 Periodic Function The periodic function is a graph with a repeating pattern. For this function we will not be finding the rule. You must know how to find the of the function. The period is the of one cycle. Example 1: Example 2: You will also be required read the graph to determine values. Example 1: In the first graph provided find f(4) You may be asked to find a value that is not on the graph Example 2: In the first graph provided find f(34) In this case you must: 1) or the period until you reach a x value that is on your graph 2) Find the value of

13 Piecewise Function The piecewise function is composed of two or more functions Ex 1: Graph Ex 2: Rule We must pay close attention to which points are and which are Reminder: On a Graph means excluded In a Rule means excluded means included means included You will be asked to the rule at different points. Example 1: Using the graph above evaluate a) f(-4) b) f(-2) c) f(0) d) f(1) Example 2 : Using the rule above evaluate a) f(-4) b) f(-3) c) f(2) d) f(5) Example 3: a) f(-2) b) f(0) c) f(2) d) f(4)

14 You may also be asked to solve for x Example: A car set off, accelerated and then travelled at a speed of 1.5 km/min for a few minutes. It then slowed down before coming to a complete stop. Function f described below represents the speed of the car according to the time elapsed from the moment it set off. 0.06x 2 if 0 x 5 f (x) = 1.5 if 5 x x + 6 if 15 x 20 where x: time elapsed, in minutes, from the moment the car set off f (x): speed of the car in km/min How much time elapsed between the two moments when the car was travelling at a speed of 0.96 km/min?