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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.
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Sections 3.3 – 3.6 Functions : Major types, Graphing, Transformations, Mathematical Models
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
Graph functions using horizontal/vertical shifts, compressions and stretches, and reflections about the x-axis or y-axis • Construct and Analyze Functions
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.
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)
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)
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??
(a) Even >> Symmetric to y-axis • (b) Neither • (c) Odd >> Symmetric to Origin
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
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)
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)
Look at Page 248 #21 • Describe increasing, decreasing, constant intervals • Increasing: (-2,0) and (2,4) • Decreasing: (-4,-2) and (0,2)
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
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.
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)
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
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
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
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
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
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.
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.
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
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
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
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
Reflection • Across the x-axis: negative multiplied outside process • Across the y-axis: negative multiplied inside process • Y =x3 • Y = -x3 • Y = (-x)3
Describe each of the following graphs. • Absolute Value • Quadratic • Cubic • Linear
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
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
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)
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.
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?
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.
Look over examples Page 278-279 • Assignment: • Pages 248, 258, 271, • Page 280 #1,7,19,27,31