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) = x 2 – 5 EVEN  G(x) = 5x 3 – x ODD  H(x) = / x / EVEN

Increasing and Decreasing Functions  Increasing: an open interval, I, if for any choice of x 1 and x 2 in I, with x 1 < x 2, we have f(x 1 ) < f(x 2 )  Decreasing: an open interval, I, if for any choice of x 1 and x 2 in I, with x 1 f(x 2 )

 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) = x 2 - 5x + 2 from 1 to 5  F(1) = 1 – = -2  F(5) = 25 – = 2  (2 – (-2)) / (5 – 1) > 4/4 > 1

Find the average rate of change for f(x) = 3x 2 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) = x 2

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) = x 2 f(x) = 3x 2 – 5x + 2  Cube: f(x) = x 3 f(x) = 2x 3 - 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 + 1if -1 < x < 1 2if x = 1 x 2 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 = x 2 y = / x /  Y = x 2 + 5y = / x / -4  Y = x 2 – 3y = / x / + 7

Horizontal Shift  Right / Left translations are caused by values added / subtracted inside the process.  +: shifts left  - : shifts right  F(x) = x 3 f(x) = x 2  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) = x 2  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 =x 3  Y = -x 3  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=x  Express the distance d from P to the origin O as a function of x. distance: sqrt [(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Sqrt[(x – 0) 2 + (x 2 – 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 [x 2 + x 4 – 2x 2 + 1] = Sqrt [x 4 – x 2 + 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(x 2 ) 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  Assignment:  Pages 248, 258, 271,  Page 280 #1,7,19,27,31