Download Precalculus

Calculus for Business
UNIT 0- Review Material: Functions and Graphs
Name:____________________________________
Date:_____________________________________
Objective:
To review prerequisite Precalculus content.
LINEAR FUNCTIONS:
Forms of Linear Equations:
slope-intercept form:
standard form:
point-slope form:
1.
Write the linear equation in slope intercept, standard, and point-slope form given the line passes
through (5, 2) and (7, 9)
2.
Write the equation of the horizontal line that passes through (-9, 2)
Intercepts
Finding x-intercepts: replace y with zero and solve
Finding y-intercepts: replace x with zero and solve
3.
a.
Find the intercepts of the following linear equations(if any):
y = 3x + 2
b.
3x + 4y = 11
4.
a.
Find the intercepts of the following non-linear equations(if any):
y=(x – 2)(x – 6)
b.
xy=9
1
Parallel & Perpendicular Lines
Parallel lines have __________ slopes.
Perpendicular lines have slopes that are ______________________
_______________________.
5.
Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to
4
the graph of y  3x 
5
6.
Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of y 
2
4
x
3
7
Intersection of Linear Functions
Elimination
1. Cancel a variable and solve
for the other.
2. Sub in result and solve for
the other variable
*Great method when neither
equation is in y=mx+b form
2x + 3y = 6
6x + 7y = 9
Methods
Substitution
1. Plug one equation in the
other and solve for the
variable
2. Sub in result and solve for
the other variable
*Great method when given only
one equation in y=mx+b form
Examples:
y=½x+3
x+y=4
Graphical
1. Graph both functions and
see where they cross.
*Great method when given two
lines in y=mx+b form
y=½x–2
y = -2x + 3
Intersection of Other Functions
Generally we stick with substitution and graphing here (may want to use separate paper for this one):
Example:
y  x 2  6x  9
x y 3
2
Calculus for Business
Lesson- Absolute Value Functions
Name:__________________________________
Date:___________________________________
Objective:
To learn how to graph absolute value functions
Do Now:
Find the intersection between y = x + 1 & y = -3x + 1
______________________________________________________________________________________
Absolute Value Functions:
Standard form: f(x) = a| x - h |+k; where ( h, k) is the vertex and “a” is the slope
if “a” is + then:
if “a” is – then:
Graph the Following
1.
f ( x)  2 x  3
2.
y
3.
y
x
4.
f ( x)   x  1  2
f ( x)  2 x  1
y
x
x
What are some things you notice about the graph of the absolute value function as a, h, k vary?
3
Calculus for Business
Lesson- Functions Rules and Notation
Name:__________________________________
Date:___________________________________
Objective:
To review basic function rules and function notation
IDENTIFYING FUNCTIONS
Definitions:
Relation:
Function:
Domain:
Range:
Examples
1.
State the domain and range of each relation, and state whether the relation is a function or not:
a.
{(-2, 0), (3, 2), (4, 5)}
b.
{(6, -2), (3, 4), (6, -6), (-3, 0)}
2.
Which relation is a function? Why? What is the shortcut rule for determining a function?
(a)
3.
(b)
(c)
(d)
Find the Domain for each:
x2  2x
x 1
a.
f ( x) 
c.
f ( x)  x  3
b.
f ( x) 
1
x 4
2
d. f(x)  - x 2  2x - 27
4
Evaluating Functions: substitute numerical value or variable into function equation and simplify
4.
find f(-1) if f(x) = x2 – 1
5.
find g(m) if g(x) = 2x6 – 10x4 – x2 + 5
6.
find k(w + 2) if k(x) = 3x + 4
7.
find h(a – 2) if h(x) = 2x2 – x + 3
Operations with Functions: given functions f and g
sum: f  g( x)  f ( x)  g( x)
product:
f  g(x)  f (x)  g(x)
difference:
f  g(x)  f (x)  g(x)
quotient:
f
f ( x)
 ( x ) 
, where g( x )  0
g( x )
 g
Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each
8.
f ( x)  5 x  4 ; g ( x)  x 2  1
9.
f ( x)  5  x ; g ( x) 
x 1
5
Composition of Functions: given functions f and g notation:  f  g ( x)  f g ( x)
Find f  g( x) and g  f ( x) for each f(x) and g(x):
10.
f ( x)  x 2  1; g ( x)  3x
11.
f ( x) 
12.
f ( x)  2 x  5 ; g ( x)  3  x
13.
f ( x)  x 2  x ; g ( x)  x  9
14.
f ( x)  2 x  3 ; g ( x)  x 2  2 x
15.
f ( x)  x  1; g ( x)  4 x 2
x  1 ; g ( x)  x  2
One-to-one Functions
A function is one-to-one when no two ordered pairs in the function have the same ordinate and different
abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test.
If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an
inverse**)
Examples: Determine whether the following functions are one-to-one.
16.
17.
f ( x)  x 2  1
f ( x)  2 x  1
18.
f ( x)  3  5 x
6
Inverse Functions
Steps:
1. Write the equation in terms of x and y.
2. Switch the x with the y.
3. Solve for y.
Examples:
19.
Find the inverse of y  4 x  8
21.
20.
Find the inverse of f ( x)  5 x  2
Graph each of the lines in examples one and two and their inverses on the same set of axes and identify
any interesting characteristics.
y
x
22.
1
Given the function f ( x )   x  2
3
(a)
Algebraically, find f -1(x).
(b)
Algebraically, verify your answer to part (a).
7
Calculus for Business
Lesson- Evaluate functions with variables;
difference quotient
Objectives:


Name:____________________________________
Date:_____________________________________
evaluate functions as expressions that involve one or more variables
explore functions by evaluating and simplifying a difference quotient
Evaluating & Simplifying a Difference Quotient:
1.
a.
For f(x) = x2 + 3x + 7, evaluate and simplify:
f x  h 
b.
f x  h   f x 
, h0
h
For each of the following, evaluate and simplify:
a.
f x  h 
b.
f x  h   f x 
, h0
h
c.
f ( x )  f (a )
, x a  0
x a
2.
f(x) = 3x2 – 2
3.
f(x) = 4x – 7
8
4.
For each given function, find and simplify:
a.
f(x) = -x2 – 2x – 4
b.
f(x) = -2x + 5
c.
f(x) = x2 + 4x + 5
f ( x  h)  f ( x )
, h0
h
9
Calculus for Business
Lesson- Symmetry, Odd/Even/Neither Functions
Name:____________________________________
Date:_____________________________________
Objectives:
~To review the algebraic method for testing for symmetry with respect to axes and origin
~To review the algebraic method for determining whether functions are even/odd/neither
~To learn shortcut approaches for the above
Odd Functions
symmetric with respect to the origin  TEST: f(-x) = -f(x)
Even Functions
symmetric with respect to the y-axis  TEST: f(x) = f(-x)
Symmetry Tests
symmetric with respect to the:
y-axis
x-axis
origin
the given equation is equivalent when:
x is replaced with -x
y is replaced with –y
x and y are replaced with –x and -y
Examples:
1.
f(x) = -x2 – 2x – 4
2.
f(x) = -2x + 5
3.
g(x) = 2x6 – 10x4 – x2 + 5
4.
f ( x)  2 x  1
10
Calculus for Business
Review of Factoring
Name:______________________________
Date:_______________________________
Objective:
Review methods of factoring including:
 Perfect square trinomials
 Difference of two squares
 Sum of two squares *
 Difference of two cubes
 Sum of two cubes
 Factoring by grouping
 Undefined terms
 Reducing to lowest terms
 Multiplying Rational Expressions
 Dividing Rational Expressions
 Adding and Subtracting Rational Expressions
 Solving Rational Equations
 Simplifying Complex Fractions
Perfect Square Trinomial
To factor a perfect square trinomial follow the steps below:
1) Create two empty binomials as indicated on the right
(
2) Take the square root of the first term of the given trinomial
3) Take the square root of the last term of the given trinomial
4) Take the result of step 2 and put in the 1st position in each binomial
5) Take the result of step 3 and put in the 2nd position in each binomial
6) The signs in the binomials should be the same as the middle term of the binomial
Example 1:
)
)(
)
factor : 4 x 2  12 x  9
Difference of Two Squares a 2  b 2
To factor a difference of two squares follow the steps below:
1) Create two empty binomials as indicated on the right
2) Take the square root of the first term of the given difference of two cubes
3) Take the square root of the last term of the given difference of two cubes
4) Take the result of step 2 and put in the 1st position in each binomial
5) Take the result of step 3 and put in the 2nd position in each binomial
6) Alternate the signs
Example 2:
)(
(
9 x 2 y 2  16
a 2  b2
Sum of Two Squares
The sum of squares is not factorable
11
Sum of Two Cubes
To factor a sum of two cubes follow the steps below:
1) Create an empty binomial and an empty trinomial as indicated on the right
(
)(
)
2) Take the cube root of the first term of the given sum of two cubes
**note- ignore all signs until the last step**
3) Take the cube root of the last term of the given sum of two cubes
4) Take the result of step 2 and put in the 1st position in the binomial
5) Take the result of step 3 and put in the 2nd position in the binomial
6) Take the result of step 4, square it and put it in the first position of the trinomial
7) Take the result of step 4 and multiply by the result of step 5 and put it in the middle position of the
trinomial
8) Take the result of step 5, square it and put it in the last position of the trinomial
9) Arrange the signs as follows ( + )( − + )
Example 3:
8 x 3  27
Difference of Two Cubes
The only difference between a difference of two cubes and a sum of two cubes is the sign arrangement.
Arrange the signs as follows ( − )( + + )
Example 4:
64 z 6  x 3
Factoring by Grouping
Steps:
1) Find a convenient point in the polynomial to partition
2) Factor within each group
3) Factor across the groups
Example 5:
Example 6:
Example 7:
16  8r  8s  r 2  2rs  s 2
Example 8.
Find the value of x that makes the expression undefined:
x 2  2 xy  y 2  z 2
2x 1
2x  4
12
Reducing- to reduce (or simplify) a rational expression means to write the answer in lowest terms. This only
applies when working with a single fraction, a product of fractions or a quotient of fractions (you cannot reduce
across a sum or difference of two or more fractions.
Example 9.
Reduce to lowest terms:
17 c 3 d 5
51c 4 d 4
If there is an addition or subtraction sign in the numerator we must factor the numerator first and then cancel
with like factors in the denominator!!!
Example 10.
Reduce to lowest terms:
x 2 y  xy2
xy
Sometimes you may also have to factor the denominator in order to reduce…
Example 11.
Reduce to lowest terms:
12  4a
a 2  a  12
Multiplying Rational Expressions:
Steps:
 Factor each numerator and denominator completely
 Cancel any like factor in any numerator with any like factor in any denominator
 Multiply the remaining expressions in each numerator
 Multiply the remaining expressions in each denominator
 Reduce if possible
Example 12.
h 2  2h  3 h 2  5h  6

h2  9
h2 1
Dividing Rational Expressions:
Steps:
 Multiply the first fraction by the reciprocal of the second (KCF)
 Factor each numerator and denominator completely
 Cancel any like factor in any numerator with any like factor in any denominator
 Multiply the remaining expressions in each numerator
 Multiply the remaining expressions in each denominator
 Reduce if possible
Example 13.
b2
1

2
b 9 3b
13
Adding and Subtracting Rational Expressions
Steps:
 Find the least common denominator among all fractions (if none already exists)
 Multiply each denominator by an appropriate factor to make it equivalent to the LCD
 Multiply each numerator by the same factor that you multiplied its denominator by
 Combine all numerators (make sure the signs are placed appropriately) and simplify and put over LCD
 Reduce if possible
Combine each set of rational expressions and simplify.
8x  4 4x  6

Example 14:
2x  6 2x  6
Example 15:
x  2y
6x  y
 2
3 x  12 y x  3 xy  4 y 2
Example 17.
1
1

5
x 1 x 1
Solving Rational Equations
Steps:
 Find the LCD
 Multiply each fraction by this LCD
 Cancel all denominators
 Solve for the variable
Solve each rational equation
2x  7 2x  9

3
Example 16.
6
10
Simplifying complex fractions
Steps:
 Find the grand LCD
 Multiply the numerator and denominator by the GLCD (will cancel out denominators within
the subfractions)
 Follow the method for reducing fractions
Example 18.
x
x2
 2
x 1 x 1
3x
2x 2

x 1 x 1
14
Calculus for Business
Lesson- Polynomial Functions including quadratics
Name:__________________________________
Date:___________________________________
Do Now:
1.
State the degree and leading coefficient of each polynomial:
a.
3b2 – 7b5 – 2b3 + 5
2.
For each polynomial equation, sketch the graph and state the number of x-intercepts. Based on your
answers, do you notice a pattern?
a.
y = 2x + 3
b.
b.
y = x2 + x – 2
6a4 + a3 – 2a
c.
y = x3 – 3x2 – x + 2
3.
Write the polynomial equation with the given roots:
Recall: Imaginary Roots Theorem: Imaginary roots occur in complex conjugate pairs.
8 and -9
3, 4i, and -4
i, -i, 5i, and -5i
1, 0, and 2i
15
4.
Write P(x) as a product of first-degree factors using the given zero (use synthetic division):
a.
P(x) = 2x3 – x2 – 26x + 40; 2 is a zero
b.
P(x) = 3x3 – 8x2 + 7x – 2; 1 is a double zero
c.
P(x) = x3 – 9x2 + x + 111; -3 is a zero
5.
Solve the following quadratic equations by completing the square:
2
1 2 3
1
2x – 5x – 12 = 0
4x2 – 16x + 20 = 0
x  x 0
2
2
4
16
Complete the square to change the following quadratic functions to the form: f ( x)  a( x  h) 2  k and
state the vertex for each parabola:
1 2
f ( x )  x  2x  3
g( x)  4x 2  12x  9
h( x)  x 2  8x  7
3
6.
Rational Roots Theorem: Possible roots 
p
, where p represents factors of the constant term and q
q
represents factors of the leading coefficient.
Process:
a. Use Rational Roots Theorem to find potential rational roots.
b. Use synthetic division, or long division, to find an actual root.
c. Repeat step 2 until the polynomial is of degree 2.
d. Factor the remaining quadratic polynomial (it is possible to get imaginary answers).
e. List all factors in simplest factored form
Find all of the factors of each:
7.
x3 – 6x2 + 25x – 150
8.
x3 – 2x2 – 23x + 60
9.
x3 – 7x + 6
10.
x3 – 11x – 10
17
Calculus for Business
Lesson- Graphing Rational Functions
Name:__________________________________
Date:___________________________________
Objective:
To review graphing a rational function
Objectives:
Graphing Rational Functions without the graphing calculator by:
a) determining zeroes of the rational function
b) determining vertical and horizontal asymptotes
c) determining slant asymptotes
d) describing the behavior of the function around vertical asymptotes
Do Now: Given the function f(x) = x3 – 6x2 + 10x – 8
(1) What is the degree of this polynomial?
(2) What is the leading coefficient?
(3) Determine if 4 is a zero of this function f(x):
Graph each (on separate graph paper) of the following functions (and check with the graphing calculator) by:
a)
b)
c)
d)
1.
f ( x) 
x
x 1
determining zeroes of the rational function
determining vertical and horizontal asymptotes
determining slant (oblique) asymptotes
describing the behavior of the function around vertical asymptotes
(increasing/decreasing)
y
x
4.
f ( x) 
5x
x 4
2
y
x
18
5.
2 x 2  3x  7
f ( x) 
x2  4
y
x
6.
x3  1
f ( x) 
x 1
y
x
19
Calculus for Business
Lesson- Graphing Piecewise Functions
Name:__________________________________
Date:___________________________________
Objective:
To review graphing a piecewise function
Do Now:
State the domain in interval notation and determine any asymptotes for the f ( x) 
x2 1
x
______________________________________________________________________________________
Piecewise functions
This is a graph that is exactly what it sounds like. It is a graph that is basically in pieces.
Graph the following:
2 x 2 if x  0
f ( x)  
2  x if x  0
y
The procedure is to graph each part of the function separately.
x
Graph the Following
2 x  3 if x  0
f ( x)  
3 if x  0
y
x
20
(1)
 4  x if x  2
f ( x)  
 x  2 if x  2
(2)
 3  x 2 if x  0
f ( x)  
 2x  3 if x  0
y
y
x
x
(3)
 3x  1
f ( x)  
2
 4x
if x  1
(4)
if x  2
x  1 if x  0
f (x)  
 1 if x  0
y
y
x
x
21
(5)
 1 if x  0

f ( x )   0 if x  0
  1 if x  0

(6)
  x  1 if x  0

f ( x)  
5 if x  0
 x  2 if x  0

y
y
x
x
(7)
 x  2 if x  1

f ( x )   2x if  1  x  1
  x if x  2

(8)
 x 2  1 if x  1
f (x)   3
 x  2 if x  1
y
y
x
x
22
Calculus for Business
Lesson- Regression and Modeling
Name:__________________________________
Date:___________________________________
Correlation and Regression
Correlation- When one variable is related to another in some way
Scatterplot- A plot on an x-y plane, where (x, y) are paired data plotted as a single point
Types of plots:
Linear Cases
Perfect +
Strong +
Moderate +
Perfect Strong Moderate _______________________________________________________________________________________
Non-Linear Cases- Only sketch the Perfect Correlations
Exponential
Quadratic
Cubic
Quartic
Natural Logarithmic
None
23
Linear Correlation Coefficient (r)
measures the strength of the linear relationship between the given variables (AKA Pearson’s Product Moment
Correlation Coefficient)
n xy   x  y 
; where n is the number of ordered pairs
r
2
2
n  x 2   x  n  y 2   y 




r 2 (in percent form) is the percent of variation in the y variable that can be explained by variation in the x
variable.
r
Type of correlation
1
Perfect +
.75  r < 1
Strong +
.50  r < .75
Moderate +
-.50 < r < .50
None
-.75 < r  -.50
Moderate -1 < r  -.75
Strong -1
Perfect Ex 1: Construct a scatterplot and compute the linear correlation coefficient between x and y.
x 1 2 3 5 9 11
y 3 5 7 10 16 20
Linear Regression
AKA- Least Squares Line, Line of Best Fit, Linear Regression Equation
This is the equation that best fits the data set given.
Calculator uses:
y = ax + b
where a = slope and b = y-intercept
slope  a 
n xy   x  y 


n  x 2   x 
2
 y  x    x  xy
y  int ercept  b 
n x    x 
2
2
2
Ex 2: Using the data from Ex 1, write the equation of the line of best fit. Plot this line on your scatterplot as
verification.
24
Using the Graphing Calculator: Note: You should only need to do steps 2-8, 10 once.
1. Enter x values in L1 and y values in L2
2. Press 2 nd y 
3. Make sure all plots are off, if not, Press 4 (Plotsoff) Enter
4. Press 2 nd y 
5. Select 1
6. Turn this plot on by highlighting On and hitting Enter
7. Press the Down Arrow once and Press Enter
8. Verify that L1 and L2 are listed.
9. Press Zoom 9 (On your graph you should see a scatterplot)
10. 2 nd 0 Press DiagnosticsOn Press Enter twice
11. Press Stat, Right Arrow, 4 (LinReg(ax + b))
12. L1 , L2 , Y1 Enter (Correlation and Regression data is now presented)
13. Zoom 9 (Line should appear on the graph verifying that equation is correct)
Making Predictions
To determine “y” given an “x” you can either:
 Sub the given “x” into the best fit model and solve for “y” or
 Enter the best fit equation into your calculator and hit: 2nd Trace “Value”. Type in the given “x” and hit
enter.
To determine “x” given a “y” you can either:
 Sub the given “y” into the best fit model and solve for “x” or
 Enter the best fit equation into your calculator. Enter the given “y” as an equation. Then find the
intersection (2nd Trace “Intersection” and follow the instructions on the screen).
Other Regressions
There are other regressions that can be determined using the graphing calculator.
Quadratic: y  ax 2  bx  c
Cubic: y  ax 3  bx 2  cx  d
Quartic: y  ax 4  bx 3  cx 2  dx  e
Natural Logarithmic: y  a  b  ln x Exponential: y  a  b x
Power Regression: y  a  x b
Sine Regression: y  A sin( Bx  C )  D
25
Examples:
1.
A study was conducted to determine if there is a relationship between the number of cookies eaten on a
daily basis and an individual male’s weight. The data are given below.
# of cookies daily
12
13
22
31
41
53
65
73
83
99
Weight in lb
120
140
170
195
205
225
250
295
310
325
a. Construct a scatterplot (complete, with appropriate labels)
b. Visually determine the best fit model.
c. Determine, using your calculator, the best regression model. Why is the one you chose the best?
Justify your answer by showing all work.
d. Determine the equation of best fit.
e. What weight would you expect a person who eats 100 cookoes per day to be?
f. How many cookies per day would you expect a person who is 150 lb to eat?
2.
The table below gives the average monthly temperature for Omaha, Nebraska, over a 30 year period.
Jan
Feb
Ma
Apr
Ma
Jun
Jul
Aug Sep
Oct
Nov Dec
21.1 26.9 38.6 51.9 62.4 72.1 76.9 74.1 65.1 53.4 39
25.1
a.
Put the data into your calculator and examine the resulting scatter plot.
b.
Use the sinreg function to regress the data and generate an equation. Plot the equation with the
scatterplot.
c.
Write the equation here:
d.
Amplitude: _______ Period: _______ Frequency: _______
26