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Comparing Functions and Graphs: Linear, Quadratic, and Exponential

In this lesson, students will compare linear, quadratic, and exponential functions and their respective graphs. They will explore average rates of change, estimating and comparing rates of change, and predicting population growth using exponential functions. The lesson also includes examples on sketching graphs given key features and comparing end behavior of polynomial and exponential functions.

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Comparing Functions and Graphs: Linear, Quadratic, and Exponential

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  1. 6-2 Comparing Functions Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

  2. Warm Up For each function, determine whether the graph opens upward or downward. downward 1. f(x) = -4x2 + 6x + 1 upward 2. f(x) = 8x2 – x - 2 Write each function in slope-intercept form. y = -3x + 10 3. Y + 3x =10 4. -6y – 12x = 24 y = -2x - 4

  3. Objectives Compare properties of two functions. Estimate and compare rates of change.

  4. The graph of the exponential function y=0.2491e0.0081x approximates the population growth in Baltimore, Maryland.

  5. The graph of the exponential function y=0.0023e0.0089x approximates the population growth in Hagerstown, Maryland. The trends can be used to predict what the population will be in the future in each city. In this lesson you will compare the graphs of linear, quadratic and exponential functions.

  6. Example 1: Comparing the Average Rates of Change of Two Functions. George tracked the cost of gas from two separate gas stations. The table shows the cost of gas for one of the stations and the graph shows the cost of gas for the second station. Compare the average rates and explain what the difference in rate of change represents.

  7. The rate of change for Gas Station A is about 3.0. The rate of change for Gas Station B is about 2.9. The rate of change is the cost per gallon for each of the Stations. The cost is less at Gas Station B.

  8. Check It Out! Example 1 John and Mike opened savings accounts on the same day. They did not deposit any money initially, but deposited each week as shown by the graph and the table. Compare the average rates of change and explain what the rates represent in this situation.

  9. Example 1 continued Mike’s average rate of change is 26. John’s average rate of change is ≈ 25.57. The rate of change is the average amount of money saved per week. In this case, Mike’s rate of change is larger than John’s, so he saves about $0.43 more than John per week

  10. Mike’s average rate of change ism = 124 -20 = 104 = 26 5-1 4 John’s average rate of change ism = 204 -25 = 179 ≈25.57 8-1 7 Example 1 continued

  11. Helpful Hint Remember to find the average rate of change over a data set, find the slope between the first and last data point. Example 1 continued

  12. The graph for the height of a diving bird above the water level, h(t), in feet after t seconds passes through the points (0, 5), (3, -1), and (5,15). Sketch a graph of the quadratic function that models the situation. Find the point that represents the minimum height of the bird. Example 2: Sketching Graphs of Functions Given Key Features.

  13. Example 2 continued Step 1 Use the points to find the values of a, b, and c in the function h(t) = at2 + bt + c. 5 = c -1 = 9a+3b+c 15=25a+5b+c

  14. Example 2 continued Step 2 Solve the system found in Step 1 and write the equation. Substitute c = 5 in 2nd and 3rd equation. -1 = 9a+3b+c 15=25a+5b+c -6 = 9a+3b 10=25a+5b 30 = -45a-15b 30=75a+15b 5 = c -1 = 9a+3b+c 15=25a+5b+c Multiply the first equation by –5 and the second equation by 3 in order to use elimination.

  15. Example 2 continued 60 = 30a Add equations and solve. 2 = a 15 = 25(2) + 5b + 5 -8 = b h(t) = 2t2 – 8t + 5

  16. minimum height: 3 ft below water level Example 2 continued Step 3 Find the minimum height of the function by finding the vertex. Graph the function and approximate the vertex.

  17. Helpful Hint Remember, in the equation f(x) = a(x - h)2 + k, the point (h, k) represents the vertex.

  18. Check It Out! Example 2 The height of a model rocket after launch is tracked in the table. Find and graph a quadratic function that describes the data.

  19. Check It Out! Example 2 continued Step 1 Use the points to find the values of a, b, and c in the function h(t) = at2 + bt + c. 31=0.25a+0.5b+c 59=2.25a+1.5b+c 55=6.25a+2.5b+c

  20. Check It Out! Example 2 continued Step 2 Solve the system found in Step 1 and write the equation. 28 = 2a+b 24=6a+2b -56 =-4a+2b 24=6a+2b 31=0.25a+0.5b+c 59=2.25a+1.5b+c 55=6.25a+2.5b+c Subtract the first equation from the second and third equations. Multiply the first equation by –2.

  21. Check It Out! Example 2 continued Add equations and solve. -32 = 2a -16 = a 28 = 2(-16) + b 28 = -32 + b 60 = b 31 = 0.25(-16) + 0.5(60) + c 31 = -4 + 30 + c 31 = 26 + c 5 = c h(t) = -16t2 + 60t + 5

  22. Check It Out! Example 2 continued Step 3 Find the maximum height of the function by finding the vertex. Graph the function and approximate the vertex. The maximum height is approximately 61 feet.

  23. Example 3: Comparing Exponential and Polynomial Functions. Compare the end behavior of the functions f(x) = -x2 and g(x) = 4 logx.

  24. The end behavior for the graph of f(x) = –x2: as x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approachesnegative infinity.The end behavior for the graph of g(x) = 4log x: as x approaches positive infinity, g(x) approaches positive infinity, as x approaches 0, g(x) → approaches negative infinity. Example 3 continued

  25. Check It Out! Example 3 Compare the end behavior of the functions f (x) = 4x2 and g(x) = x3.

  26. The end behavior for the graph of f(x)= 4x2: as x approaches positive infinity, f(x) approaches positive infinity. As x approaches negative infinity, f(x)approaches positive infinity. The end behavior for the graph of g(x) = x3: as x approaches positive infinity, g(x)approaches positive infinity, as x approaches negative infinity, g(x) approaches negative infinity. Check It Out! Example 3 continued

  27. Lesson Quiz: Part I Compare the end behavior of each pair of functions. 1. f(x) = x and g(x) = -x4 f(x): as x approaches positive infinity, f(x) approaches positive infinity; as x approaches negative infinity, f(x) approaches negative infinity. g(x): as x approaches positive infinity, g(x) approaches negative infinity; as x approaches negative infinity, g(x) approaches negative infinity.

  28. Lesson Quiz: Part 2 2. f(x) = 4ex and g(x) = log x f(x): as x → ∞, f(x) → ∞; as x→ –∞, f(x) → 0. g(x): as x →∞, g(x) → 1; as x → 0, g(x)→ –∞. 3. Find the equation of a quadratic function that describes the data in the table. f(x) = 3x2 -4x -10.

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