Types of Functions Here are six common types of functions and examples of each. Linear Quadratic Absolute Value Square Root Exponential Reciprocal

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1 Topic 2.0 Review Concepts What are non linear equations? Student Notes Unit 2 Non linear Equations Types of Functions Here are six common types of functions and examples of each. Linear Quadratic Absolute Value Square Root Exponential Reciprocal 2 1 y 2x 1 y x 4x 3 y x 1 y x 1 2 y 2 x y x We will be focusing on two specific types of non linear graphs. Quadratic. 2 y ax bx c Exponential. y ab x Given the following equations classify them as linear or Non linear. a. 2x 3y 7 b. 2 2 x y 16 c. x 2y 8 d. y x 4 e. y 3 x 1 y 3 4 x

2 Topic 2.1 Modeling Exponential Data. Example 1 The population of a city was growing at a rate of 5% annually. In 1990, the population was a) Determine the ratio the population grows by and create a table of values for the annual population of the city during the years Year Population b) determine the equation by using the Exponential Regression function of your graphing calculator. c) Select appropriate window settings so that the graph will show data for years and write them down below. Draw a sketch of the graph below. Window Settings X = (,, ) Y = (,, ) d) Estimate the year when the population exceeds using your regression equation

3 Example 2 A colony of insects doubles in size every 30 days. The initial population of insects is 500. a. determine the size of the colony after 75 days. b. Determine when the colony will reach a size of 100,000 insects. Complete the following table for the number of insects after each time period. Time in Number of Days Insects (x) (y) Sketch your graph and record your window settings. Let Ymin=0. The rest of the settings should be appropriate for your data. Xmin= Xmax= Xscl= Ymin=

4 Radioactive Decay 1. The half-life of sodium-24 is approximately 15 hours. a) Determine an exponential decay equation for sodium-24 that models the percent of sodium-24 remaining. Use the regression feature on the calculator. (Because we are working with percent, assume there is 100 g of sodium-24 initially.) Time (h) Mass Remaining (g) b) Sketch the graph of the coordinates and the regression line, correctly labeling the axes. c) Determine the percent of a sample of sodium-24 that remains after 24 h. d) Calculate when there would be 15 grams of sodium-24 remaining.

5 2. A strain of bacteria doubles in number every 8 hours. There are initially 200 bacteria in a sample. a) Graph the equation up to 3 days. b) Write and equation that represents the growth of the bacteria. c) Determine the number of bacteria after 11 hours.. d) Determine the number of bacteria after 1.5 days..

6 3. The half life of Polonium 210 is approximately 138 days b) Graph the equation up to 3 years. b) Determine an exponential decay equation for Polonium that models the percent of Polonium remaining(hint- initial amount is 100% mass). c) Determine the percent of a sample of Polonium that remains after 48 days. d) Determine the percent of a sample of Polonium that remains after 3.5years.

7 Exponential Regression If you have a set of data that you think shows an exponential growth pattern, you can use a graphing calculator to test your hypothesis. 1. Press the STAT key, the 5 key and the ENTER key to select SetUpEditor. 2. Press the STAT key and the 1 key to display the lists. 3. Clear the values from any previous lists by moving the cursor to the header of each line and pressing CLEAR followed by ENTER. 4. Enter the data into the lists by pressing the ENTER key after each item. Note: The first list works best if it starts at 0 and increases accordingly. 5. Press the STAT key, the right arrow key and the 0 key to select ExpReg. 6. Press the L1 key, the comma key, the L2 key, the comma key, t he VARS key, the right arrow key, the 1 key, the 1 key again and the ENTER key to define the regression as x-values from L1, y values from L2 and plotting Y1. 7. Press the Y= key to see the equation displayed and the GRAPH key to see the graph. 8. You can now use the other features of the calculator to determine points on the graph: TRACE will move the cursor along the curve WINDOW will allow you to change the size of the viewing window CALC will allow you to calculate values of y for given values of x.

8 Topic 2.2 Modeling Quadratic Data Quadratic Graphs Analyze Domain, Range, Intercepts, Max/ Min, Vertex, Axis of Symmetry. Analyze the equation of a quadratic. Quadratic Regressions. Quadratic Regression 1. Press the STAT key, the 5 key and the ENTER key to select SetUpEditor. 2. Press the STAT key and the 1 key to display the lists. 3. Clear the values from any previous lists by moving the cursor to the header of each line and pressing CLEAR followed by ENTER. 4. Enter the data into the lists by pressing the ENTER key after each item. 5. Press the STAT key, the right arrow key and the 5 key to select QuadReg. 6. Press the L1 key, the comma key, the L2 key, the comma key, t he VARS key, the right arrow key, the 1 key, the 1 key again and the ENTER key to define the regression as x-values from L1, y values from L2 and plotting Y1. 7. Press the Y= key to see the equation displayed and the GRAPH key to see the graph. 8. You can now use the other features of the calculator to determine points on the graph: TRACE will move the cursor along the curve WINDOW will allow you to change the size of the viewing window CALC will allow you to calculate values of y for given values of x.

9 2.2 Problem Solving Using Quadratic Functions Introduction: Have you ever wanted to attend a concert or game but found that the tickets were too expensive? The arena or concert hall manager tries to determine the ticket price that will maximize revenue. If she sets the price too low, revenue will be low even if all the tickets are sold. If she sets the price too high, people will not buy the tickets and, again, revenue will be low. The height of a projectile, h metres above the ground, at time t seconds, is given by the equation h = 1 gt 2 + vt + s 2 where g is the acceleration due to gravity (9.8 m/s 2 on Earth); v is the initial vertical velocity in metres per second; and s is the initial height above the ground in metres. 1. Annie Pelletier won a bronze medal in the women s springboard competition at the 1996 Summer Olympics in Atlanta. Pelletier somersaults from a 3-m springboard, with an initial upward velocity of 8.8 m/s. a) Use the general formula given to determine the equation that models the height above the water, h metres, at time t seconds after she leaves the diving board. b) Graph this equation on a calculator. Sketch it and indicate appropriate window settings. x: [,, ] y: [,, ] c) Use this graph to determine Annie s maximum height, and the time taken to reach this height. d) Use the graph to determine the time Annie is in the air.

10 2. An arena or concert hall manager tries to determine the ticket price that will maximize revenue. If the price is too low, revenue will be low even if all the tickets are sold. If the price is too high, people will not buy the tickets and the revenue will still be low. A hockey arena has 1600 seats. When the price is $10, all the seats are sold. The manager wants to increase revenue. The following survey shows the results. Ticket Price ($) Expected Ticket Sales a. Draw a graph to illustrate the expected revenue for the different ticket prices. b. Use the graph to determine the price that will produce the maximum revenue. a. Ticket Price ($) Revenue ($) Use the window settings: {0, 30, 2} {0, 22000, 2000} b.

11 So far, we have graphed parabolas comparing height above the ground to time. These parabolas are not trajectories. A trajectory is the actual path of the object moving through the air. It relates the height above the ground to the horizontal distance traveled, and it does illustrate the path of a moving object. (i.e. the soccer question) 3. A soccer ball reaches a maximum height of 2.3 m before landing 31.8 m away. Assume that its trajectory is a parabola. a) List three points on the trajectory that can be plotted on the calculator. Sketch the points. b) Determine the equation of the parabola that models the trajectory. c) Sketch the graph. d) Determine the domain and range of the parabola.

12 4. A rock is dropped from a bridge into a river. Its height, h meters, above the river t seconds after it is released is modeled by the quadratic function h(t) = t 2. a) Graph the function for reasonable values of t. Label your axes and include your window choices below. Windows x[,, ] y[,, ] b) How tall is the bridge? c) How high is the rock after 2.5 seconds? d) State the domain and range of the function (for this situation).

13 5. You are standing on a cliff, 110 m above a beach. You throw a stone straight up at 17 m/s. After reaching its maximum height, the stone falls to the beach just missing the cliff. The equation of the path of the stone is given by the equation h(t) = -4.9t t a) Graph the function correctly labeling the axes. Include your window choices. Windows x[,, ] y[,, ] b) Determine the height of the stone after each time. (2 nd calc value) i) 1.0 s ii) 2.4 s iii) 4.5 s iv) 6.1 s c) At what time(s) is the stone at each height? (make another equation y = and find the intersection point. i) 110 m ii) 90 m iii) 60 m iv) 30 m d) What is the maximum height of the stone? How long does it take the stone to reach this height? e) How long does it take the stone to hit the beach? f) What is the domain and range of this function?

14 Topic 2.3 Graphing Quadratics Ezample I. For each of the following functions: complete table values, sketch the graph, use your calculator to determine the x-intercepts and y-intercept, find the vertex and axis of symmetry. Describe the domain and the range. 1. y = x x intercepts: y intercept: vertex: axis of symmetry: Domain: Range: 2. y = x 2 + x 6 x intercepts: y intercept: vertex: axis of symmetry: Domain: Range:

15 3. y = 4x 2 8x x intercepts: y intercept: vertex: axis of symmetry: Domain: Range: 4. y = 2x 2 + 8x + 3 x intercepts: y intercept: vertex: axis of symmetry: Domain: Range:

16 Topic 2.4 Solving Non-Linear Equations Using a Graphing Calculator To solve an equation means to fine the value(s) of x that make the equation true. Ex. Solve 0 = 2x + 8 Check The solution to the equation is also called a zero. When y = 0, what is x, the solution? 0 = 2x + 8 is an equation y = 2x + 8 is a function We graph a function and solve an equation. As you can see, they are similar (but different). Graph the function y = 2x + 8 on your graphing calculator. Find the solution using your graphing calculator. As you can see, you don t always have to use algebra to solve an equation. We can use two different methods for solving a non-linear equation too.

17 Investigation: Method Graph the function y = x 2 2x 5. Sketch your graph. On the graph, where is y < 0? On the graph, where is y > 0? Where is y = 0? What is x when y = 0? The solution(s) to what is x when y = 0 is the solution to the equation x 2 2x 5 = 0 2. Explain how we found the solution to the equation x 2 2x 5 = 0 using method 1.

18 Let s manipulate the equation x 2 2x 5 = 0 We could have: Method 2 or 1. Graph the two functions y = x 2 and y = 2x + 5 on the same screen. These functions form a non-linear system. A system of equations is a set of two or more equations that are examined at the same time. Sketch your results here. 2. Locate all the intersection points. How many are there? Are there always this many? Intersection points: 3. How are these intersection points similar to the zeros found using method 1? How are these intersection points different from the zeros found using method 2? What are we trying to solve for when we solve an equation?

19 Examples. Solve using two methods. 1. x 2 = 5x 4 Method 1 Method 2 (Graph one function and find the zeros) (Graph two functions and find the intersection points) 2. 4x + 5 = 2 x Method 1 Method 2 (Graph one function and find the zeros) (Graph two functions and find the intersection points) 3. x 3 + x 20 = 0 Method 1 Method 2 (Graph one function and find the zeros) (Graph two functions and find the intersection points)