1 / 35

Graphing and Analyzing Functions: Symmetry, Increasing/Decreasing Intervals, Local Extremas, and Average Rate of Change

This section explores determining even and odd functions, using graphs to find increasing and decreasing intervals, identifying local extrema, and calculating the average rate of change of a function. It also covers graphing common functions and understanding their properties.

dwestcott
Download Presentation

Graphing and Analyzing Functions: Symmetry, Increasing/Decreasing Intervals, Local Extremas, and Average Rate of Change

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Sections 3.3 – 3.6 Functions : Major types, Graphing, Transformations, Mathematical Models

  2. Objectives for Class • Determine even and odd functions • Use a graph to determine increasing and decreasing intervals • Identify local maxima and minima • Find the average rate of change of a function • Graph common functions, including piece-wise functions

  3. Graph functions using horizontal/vertical shifts, compressions and stretches, and reflections about the x-axis or y-axis • Construct and Analyze Functions

  4. Properties of Functions • Intercepts: Y-intercept: value/s of y when x=0 to find substitute a 0 in for x and solve for y x –intercept: value/s of x when y=0 x-intercept/s are often referred to as the “ZEROS” of the function to find substitute a 0 in for y and solve for x Graphically these occur where the graph crosses the axes.

  5. Even and Odd Functions • Describes the symmetry of a graph • Even: If and only if whenever the point (x,y) is on the graph of f, then the point (-x,y) is also on the graph. f(-x) = f(x) >>Symmetry Test for y-axis If you substitute a –x in for x and end up with the same original function the function is even (symmetric to y-axis)

  6. Odd: If and only if whenever the point (x,y) is on the graph of f, then the point (-x,-y) is also on the graph. f(-x) = -f(x) >>>correlates with symmetry to the origin If you substitute a –x in for x and get the exact opposite function the function is odd (symmetric to the origin)

  7. Theorem • A function is even if and only if its graph is symmetric with respect to the y-axis. A function is odd if and only if its graph is symmetric with respect to the origin. • Look at the diagrams on the bottom of page 241 >>Odd or Even or Neither??

  8. (a) Even >> Symmetric to y-axis • (b) Neither • (c) Odd >> Symmetric to Origin

  9. Determine if each of the following are even, odd, or neither • F(x) = x2 – 5 • EVEN • G(x) = 5x3 – x • ODD • H(x) = / x / • EVEN

  10. Increasing and Decreasing Functions • Increasing: an open interval, I, if for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2) • Decreasing: an open interval, I, if for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2)

  11. Constant: an interval I, if for all choices of x in I, the values f(x) are equal. • Look at diagram on page 242 • Increasing intervals ( -4,0) • Decreasing intervals (-6,-4) and (3,6) • Constant interval (0,4)

  12. Look at Page 248 #21 • Describe increasing, decreasing, constant intervals • Increasing: (-2,0) and (2,4) • Decreasing: (-4,-2) and (0,2)

  13. Local Maxima / Minima • Maxima: highest value in one area of the curve A function f has a local maximum at c if there is an open interval I containing c so that, for all x does not equal c in I, f(x) < f(c). We call f(c) a local maximum of f. • Minima: lowest value in one area of the curve

  14. A function f has a local maximum at c if there is an open interval I containing c so that, for all x does not equal c in I, f(x) > f(c). We call f(c) a local maximum of f. A local maximum is a high value for all values around it.

  15. Find the local maxima/minima for the function on page 244 and #21, page 248 • Page 244 • Local Maxima: (1,2) • Local Minima: (-1,1) and (3,0) • #21, Page 248 • Local Maxima: (-4,2), (0,3), (4,2) • Local Minima: (-2,0), (2,0)

  16. Average Rate of Change • Formula: Change in y / Change in x • Example: Find average rate of change for f(x) = x2 - 5x + 2 from 1 to 5 • F(1) = 1 – 5 + 2 = -2 • F(5) = 25 – 25 + 2 = 2 • (2 – (-2)) / (5 – 1) > 4/4 > 1

  17. Find the average rate of change for f(x) = 3x2 from 1 to 7 • F(1) = 3 • F(7) = 147 • (147 – 3) / (7 – 1) • 144 / 6 > 24

  18. Major Functions • Graph each of the following on the graphic calculator. Determine if each is even, odd, or neither. State whether each is symmetric to the x-axis, y-axis, or origin. State any increasing/decreasing intervals. (Draw a sketch of the general shape for each graph.) F(x) = cube root of x F(x) = / x / F(x) = x2

  19. Library of Functions: look at the shape of each • Linear: f(x) = mx + b 2x + 3y = 4 • Constant: f(x) = g f(x) = 4 • Identity: f(x) = x f(x) = x • Quadratic: f(x) = x2 f(x) = 3x2 – 5x + 2 • Cube: f(x) = x3 f(x) = 2x3 - 2

  20. More Functions • Square Root: f(x) = square root of x f(x) = square root of (x + 1) • Cube Root: f(x) = cube root of x f(x) = cube root of (2x + 3) • Reciprocal Function: f(x) = 1/x f(x) = 3 / (x + 1) • Absolute Value Function: f(x) = / x / f(x) = 2 / x + 1 / • Greatest Integer Function: f(x) = int (x) = [[x]] greatest integer less than or equal to x f(x) = 3 int x

  21. Piecewise Functions • One function described by a variety of formulas for specific domains • F(x) = -x + 1 if -1 < x < 1 2 if x = 1 x2 if x > 1 Find f(0), f(1), f(4) Describe the domain and range.

  22. Application • A trucking company transports goods between Chicago and New York, a distance of 96o miles. The company’s policy is to charge, for each pound, $0.50 per mile for the first 100 miles, $0.40 per mile for the next 300 miles, $0.25 per mile for the next 400 miles, and no charge for the remaining 160 miles. • Find the cost as a function of mileage for hauls between 100 and 400 miles from Chicago. • Find the cost as a function of mileage for hauls between 400 and 800 miles from Chicago.

  23. Transformations • Vertical Shifts: values added/subtracted after the process cause vertical shifts • +: up • - : down • Y = x2 y = / x / • Y = x2 + 5 y = / x / -4 • Y = x2 – 3 y = / x / + 7

  24. Horizontal Shift • Right / Left translations are caused by values added / subtracted inside the process. • +: shifts left • - : shifts right • F(x) = x3 f(x) = x2 • F(x) = (x – 2)3 f(x) = (x + 1)2 • F(x) = (x + 5)3 f(x) = (x -6)2

  25. Compressions and Stretches • Coefficients multiplied times the process cause compressions and stretches • F(x) = / x / • F(x) = 2 / x / • F(x) = ½ / x / • /a/ > 1 : stretch • /a/ < 1: compression

  26. Horizontal Stretch or Compression • Value multiplied inside of process • F(x) = x2 • F(x) = (3x)2 • F(x) = (1/3x)2 • /a/ > 1: horizontal compression • /a/ < 1: horizontal stretch

  27. Reflection • Across the x-axis: negative multiplied outside process • Across the y-axis: negative multiplied inside process • Y =x3 • Y = -x3 • Y = (-x)3

  28. Describe each of the following graphs. • Absolute Value • Quadratic • Cubic • Linear

  29. Describe the transformations on the following graph • F(x) = -4 (square root of (x – 1)) • - : reflection across x axis • 4: vertical stretch • -1 inside process: 1 unit to right

  30. Write an absolute value function with the following transformations • Shift up 2 units • Reflect about the y-axis • Shift left 3 units • F(x) = /-(x + 3)/ + 2

  31. The perimeter of a rectangle is 50 feet. Express its area A as a function of the length, l, of a side. • l + w + l + w = 50 • 2l + 2w = 50 • l + w = 25 • W = 25 – l • A(l) = lw = l(25 – l)

  32. Let P = (x,y) be a point on the graph of y=x2 - 1 • Express the distance d from P to the origin O as a function of x. distance: sqrt [(x2 – x1)2 + (y2 – y1)2 Sqrt[(x – 0)2 + (x2 – 1)2] • What is d if x = 0? • Sqrt [(0 – 0)2 + (0 – 1)2] = 1 • What is d if x = 1? • Sqrt [(1 – 0)2 + (1 – 1)2] = 1 • Distance from curve to origin? • Sqrt [x2 + x4 – 2x2 + 1] = Sqrt [x4 – x2 + 1] • Plug x values into equation formed to find d.

  33. See example page 277 • A rectangular swimming pool 20 meters long and 10 meters wide is 4 meters deep at one end and 1 meter deep at the other. Water is being pumped into the pool to a height of 3 meters at the deep end. • Find a function that expresses the volume of water in the pool as a function of the height of the water at the deep end. • Find the volume when the height is 1 meter? 2 meters? • Use a graphing utility to graph the function. At what height is the volume 20 cubic meters?

  34. Let L denote the distance (in meters) measured at water level from the deep end to the short end. L and x (the depth of the water) form the sides of a triangle that is similar to the triangle with sides 20 m by 3 m. • L / x = 20 / 3 • L = 20x / 3 • V = (cross-sectional triangular area) x width = (½ L x)(10) = ½ (20/3)(x)(x)(10) = 100/3(x2) cubic meters. • Substitute 1 in to find volume when height is 1 meter. • Substitute 2 in to find volume when height is 2 meter. • Graph and trace to find l when volume is 20 cubic meters.

  35. Look over examples Page 278-279 • Assignment: • Pages 248, 258, 271, • Page 280 #1,7,19,27,31

More Related