Sections 3.3 – 3.6Functions : 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=0to find substitute a 0 in for x and solve for y

x –intercept: value/s of x when y=0x-intercept/s are often referred to as the “ZEROS” of the functionto 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

curveA 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 xF(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 xf(x) = 3 int x

Piecewise Functions One function described by a variety of formulas for

specific domains

F(x) = -x + 1 if -1 < x < 12 if x = 1x2 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