Pre-Calculus
Chapter 1
Name_________________________________________________
Date
Day
Lesson
10/4
Thursday
1.1
10/8
Monday
1.2
10/10 Wednesday
1.3
10/12
Friday
Quiz 1.1-1.3
10/16
Tuesday
1.4
10/22
Monday
1.5
10/24 Wednesday
1.6
10/26
Friday
Review
10/30
Tuesday
Test
Notes:
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Pre Calculus Chapter 1
Assigned
p.11 (vocab: 1-5; 3, 4, 6-44
even, 45, 46, 54, 56, 57, 75,
113-116)
p. 24 (vocab: 1-3, 5; 2-8 even,
15-42 3rds, 53-61odds, 78, 8790, 101-104)
p. 38 (vocab: 1-6; 3-54 3rds, 6066 3rds, 95-100)
rds
p. 48 (vocab: 1, 2, 6; 3-9 3 , 15rds
20, 21-42 3 , 59, 60. 69-74)
p. 58 (vocab: 1-4; 3-24 3rds, 3654 3rds, 97-100)
p. 69 (vocab: 1-5; 3-18 3rds, 2934, 36-78 6ths, 106, 115, 122)
Review
Extra Credit
TEST
Due
2
Pre Calculus Chapter 1
1.1 Lines in the Plane
In this section you will learn to graph an equation by analytical methods and by using the help of a graphing
calculator. Then you will study lines and their equations. Remember a line is a straight line.
Sketching the Graph of an Equation by Point Plotting
Example 1: Use the point plotting method to sketch the graph of 𝑦 = 1 − 𝑥 2 . Make a table of values first,
sketch the graph and then check your graph with a graphing calculator.
(Choose at least five x-values at random and then find the corresponding y-values.)
x
-3
-2
-1
0
2
3
𝑦 = 1 − 𝑥2
Example 2: Use your calculator and a good viewing window to graph 2𝑦 + 𝑥 3 = 4𝑥. First solve the equation
for y in terms of x.
Select the best viewing window for the equation:
A. [-3, 3] by [-9, 0]
B. [0, 5] by [-10, 10]
C. [-5, 5] by [-10, 10]
10𝑥
Example 3: Use your calculator to graph the function 𝑦 = 𝑥 2 +1
How many times does the graph cross the x-axis? _____
Standard viewing window used [-10, 10] by [-10, 10]
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Pre Calculus Chapter 1
Example 4: A rectangle of length x and width w has a perimeter of 12 inches.
a. Show that the area of the rectangle is 𝑦 = 𝑥(6 − 𝑥). (draw a picture)
b. Graph the area equation from above.
c. Graph only the portion of the graph that makes sense for this situation. (area can’t be negative)
d. What is the largest area when the perimeter is 12 inches?
e. What dimensions give the largest area?
Lines in the Plane
Slope
Slope represents the steepness of a line. (Slope can be represented in several different notations.) The slope
m of a line passing through the points (x1 , y1) and
( x2 , y2) is
m =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
Δ𝑦
Δ𝑥
=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
Example 5: Find the slope between the given points
a. (-2, 0) (3, 1)
b. (-1, 2) (2, 2)
c. (0 ,4) (1 , -1)
d. (3, 4) (3, -8)
If the slope of a line:
1. Has a positive slope, m>0, the line rises from left to right.
2. Has a negative slope, m<0, the line falls from left to right
3. Has zero slope �
0
�, m=0, the line is horizontal.
𝑎𝑛𝑦 #
4. Has undefined slope �
Do the slopes in ex. 5 rise or fall? A.
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Pre Calculus Chapter 1
B.
𝑎𝑛𝑦 #
0
�, the line is vertical.
C.
D.
Example 6: Estimate the slopes of the following lines, remember to read graphs from left to right.
m=
m=
The following are equations of lines. They are helpful in either sketching graphs of lines or finding equations
of lines when given a graph.
Forms of linear equations:
1. General Form: 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
2. Slope-intercept: 𝑦 = 𝑚𝑥 + 𝑏
3. Point-Slope form: 𝑦 – 𝑦1 = 𝑚(𝑥 – 𝑥1 )
4. Intercept form:
𝑥
𝑎
𝑦
+ = 1 , 𝑎 ≠ 0, 𝑏 ≠ 0
𝑏
5. Vertical Line: 𝑥 = 𝑎
6. Horizontal Line: 𝑦 = 𝑏
Don’t forget  Slope formula: 𝑚 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Example 7: Find equations given the following information
a. (1, -2) m = 3, in slope-intercept form
b. (2, 1) m = 0, in point-slope form
c. (6, 2) & (7, 0), in general form
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Pre Calculus Chapter 1
Example 8: Given the following equations, find the slope and y-intercept (if possible) of the equation of the
line.
a. 5𝑥 − 𝑦 + 3 = 0
b. 3𝑦 + 5 = 0
c. 𝑥 = −4
Sketching graphs of a line – easiest form to graph a line is from the Slope-Intercept form of a line.
1st- start at the y-intercept (on the y-axis)
2nd- count off the slope in both directions
Now… sketch the lines from example 8
a.
b.
c.
Example 9: Application
A teacher’s salary was $32,000 in 2004 and $34,200 in 2006. The teacher’s salary follows a linear growth
pattern. What should the teacher’s salary be in 2008?
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Pre Calculus Chapter 1
Parallel lines: Parallel lines run side by side but never touch. Their slopes are __________________.
Notation: _____________________
Perpendicular lines: perpendicular lines intersect each other at a 90° angle. There slopes are
_____________________
_____________________ of each other. Notation:_________
Example 10: Determine whether the lines L1 and L2 passing through the pairs of points are parallel,
perpendicular or neither.
L1: (0, -1) (5, 9)
L2: (0, 3) (4, 1)
Example 11: Find an equation of the line that passes through the point (-1, 0) and is parallel to 𝑦 = −3
Review Topic:
Factor the following trinomials:
1. 𝐱 𝟐 + 𝟖𝐱 + 𝟏𝟐
2. 𝟔𝐱 𝟐 − 𝟑𝟐𝐱 + 𝟏𝟎
3. What is a polynomial? What is the standard form of a polynomial?
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Pre Calculus Chapter 1
Extra Examples:
1. Find the slope intercept form of a line that passes through the point (2, -1) and is a) parallel to and b)
perpendicular to the given line. *Give solution in slope intercept form.
• 2𝑥 – 3𝑦 = 5 (a)
(b)
2. Write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the
given line and (b) perpendicular to the given line.
• (-3, 2) x + y = 7
(a)
(b)
3. You are given the dollar value of a product in 2006 and the rate at which the value of the product is
expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the
product in terms of the year t. (Let t = 6 represent 2006)
• $156 $4.50 increase per year
4. Find the general form of the equation of the line that passes through the given point and has the indicated
slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.
• (-3, 6) m = -2
•
(6, 1)
8
m is undefined
Pre Calculus Chapter 1
1.2 Functions
In this section, you will learn about different functions and how to find domain.
Any set of points in the coordinate plane is called a relation. Here are two examples:
1. The simple interest I earned on an investment of $1000 for 1 year is related to the annual interest rate
r by the formula 𝐼 = 1000𝑟
2. The area A of a circle is related to its radius r by the formula 𝐴 = 𝜋𝑟 2
Definition of a Function
A function f from a set A to a set B is a rule of correspondence that assigns to each element x in a set A
exactly one element in the set B. The set A is the domain (or set of inputs) of the function f, and the set B
contains the range (or the set of outputs).
For an x-y relation to be a function of x, each x value must produce only 1 y value. If y is a function of x, such
as 𝑦 = 𝑥 2 − 2𝑥 + 3, then x is known as the independent variable and y is known as the dependent variable.
To each value of the independent variable there corresponds exactly one value of the dependent variable.
The set of all values which can be used for the independent variable is called the domain of the function. The
set of all resulting values for the dependent variable is known as the range of the function.
•
Look at the function that relates the time of day to the temperature.
What is the input and what is the output?
Characteristics of a Function from Set A to Set B
1.
2.
3.
4.
Each element in A must be matched with an element in B
Some elements in B may not be matched with any element in A
Two or more elements of A may be matched with the same element of B
An element of A (domain) cannot be matched with 2 different elements of B
Example 1: Think of other real life situations where it is important for the relation to be a function.
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Pre Calculus Chapter 1
There are some different ways of writing or identifying functions
1. Verbally: (writing as a sentence)
The input value x is the election year from 1952 to 2004 and the output value y is the President of the
United States.
2. Numerically:
• Ordered pairs – (1:00, 89), (2:00, 90), (3:00, 93), (4:00, 88)
• Mapping –
•
Table Input, x
Output, y
-3
-9
-1
-1
0
0
1
1
3
9
3. Graphically: ordered pairs are graphed on a coordinate plane.
4. Algebraically: function is given as an equation in 2 variables. 𝑦 = 2𝑥 3 + 5
Example 2: Determine which of the sets of ordered pairs represents a function from A to B. Give reasons for
your answers.
A = {0, 1, 2, 3} and B = {-2, -1, 0, 1, 2}
1. { (0,1), (1, -2), (2, 0), (3, 2) }
Reason: _________________________________________________________
2. { (2, 2), (1, -2), (3, 0), (1, 1) }
Reason: _________________________________________________________
3. { (0, 0), (1, 0), (2, 0), (3, 0) }
Reason: _________________________________________________________
4. { (0, 2), (3, 0), (1, 1) }
Reason: _________________________________________________________
Example 3: State whether the equations determines y as a function of x.
A. 𝑥 2 + 𝑦 2 = 4
B. 𝑥 2 + 𝑦 = 4
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Pre Calculus Chapter 1
Functions are often written in what is known as function notation: 𝑦 = 𝑓(𝑥). Instead of 𝑦 = 𝑥 2 , you may see
𝑓(𝑥) = 𝑥 2 . They mean the same thing. Learn how to work with function notation.
A function value such as f(3) means the value of f(x) (or the value of y) when 𝑥 = 3.
Example 4a: For 𝑓(𝑥) = 𝑥 2 find the following:
𝑓(0) = _________ = ______
𝑓(3) = __________ = ______
𝑓(−1) = _________ = ______
𝑓(𝑥 + 2) = ___________ = _____________
𝑓(1) = ____________ = _______
𝑓(𝑝) = _________ = ______
Example 4b: For 𝑓(𝑥) = 2𝑥 2 − 3𝑥 + 5 find the following:
𝑓(−1) = _________ = ______
𝑓(𝑥 − 1) = ___________ = _________________
Example 5: For the function given by 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 7, find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
𝑓(𝑥 2 ) __________=________
(this is called a difference quotient)
Piecewise Functions
A piecewise defined function is a function that is defined by two or more
equations over a specified domain.
One example of a piecewise function is:
𝑓(𝑥) = �
𝑥 2 + 1,
𝑥 − 1,
𝑥<0
𝑥≥0
Example 6: Evaluate the above function at x = -1, 0, 1. (hint: find the piece which contains the specified xvalues)
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Pre Calculus Chapter 1
One of the real challenges in working with functions is determining the domain and range for a given function.
Sometimes a domain explicitly described, such as: domain is 0 ≤ 𝑥 ≤ 2 or [0, 2]. In such cases, you would
need only to find a corresponding range.
However, usually the domain is not described, it is implied. In such cases the domain consists of all real
number for which the function is defined.
It is not always a simple task to find restrictions on the domain (values of x that would cause the function to be
undefined). It is even a more difficult challenge to find the range of a function. Practice helps, but reasoning
skills are indispensable here.
*Always be on the lookout for restrictions on the domains of functions.
Two of the more important types of restrictions on the domain are:
Watch for:
1. Values which make
zero. To find these, set the denominator = 0
2. Values which make
(expressions under the radical sign) with even nth roots negative. (square
roots, 4th roots, 6th roots, etc)  To find these values set the radicand ≥0 and solve
*REMEMBER: negative exponents can be fractions in disguise, and fractional exponents can be roots in
disguise!
Example 7: Find the domain of the following functions, express it in interval or set notation. Justify your
answer using algebraic methods. Check your answer using your calculator.
𝑥+1
a. 𝑓(𝑥) = 𝑥−4
b. 𝑓(𝑥) = √2𝑥 − 5
c. 𝑓: {(−3, 0), (−1, 4), (0, 2), (2, 2), (4, −1)}
d. 𝑓(𝑥) = 𝑥 2 − 𝑥 + 6
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Pre Calculus Chapter 1
Visually finding the domain
Given a graph, the domain are the x-values the graph covers from left to right.
The range is the y-values the graph covers up and down.
Find the domain and range of the graph at the right.
Write the solution in interval notation.
Applications
Example 8: A balloon ascends vertically from a point 20 feet from a house. Let d be the distance between the
balloon and the house. Express the height of the balloon as a function of d. What is the domain of the
function?
d
h
house
20
Example 9: A company invests $98,000 for equipment to produce a product. Each unit of the product costs
$12.30 and is sold for $17.98. Let x be the number of units produced and sold.
a. Write the total cost C as a function of x.
b. Write the revenue R as a function of x.
c. Write the profit P as a function of x. (Note: P = R – C)
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Pre Calculus Chapter 1
Review Topic:
Perform the operation and simplify.
1.
2.
3𝑥
𝑥 2 −𝑥−6
5
+ 𝑥 2 +2𝑥
2𝑥 2 −3𝑥−9 4𝑥+6
∙
2𝑥
𝑥−3
Extra Examples:
Decide if each is a function –
A.
x
2
2
3
4
5
y
-9
-1
0
1
9
B.
C.
D. { (0, 0), (1, 0), (2, 0), (3, 0) }
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Pre Calculus Chapter 1
1.3 Graphs of Functions
In this section, you will study functions from a geometric perspective.
***There are many special “Technology Tips” on pages 32, 33, and 35 of the book. These may help you to
understand your graphing calculator better.
*Remember*
Domain: The ________________ on the graph.
Range: The_________________ on the graph.
Example 1: Use the graph of the function to find the domain and range of f. Then find f(0).
** This is a __________________ function
Example 2: Use a graphing utility to graph the function and estimate its domain and range. Then find the
domain algebraically. Sketch the graph.
𝑓(𝑥) = √𝑥 − 1
By the definition of a function, at most one y-value corresponds to a given x-value. So, a vertical line can
intersect the graph of a function at most once.
Vertical Line Test: A set of points in a coordinate plane is the graph of y as
a function of x if and only if no vertical line intersects the graph at more
than one point.
Drag your pencil vertically across the graph and if at any time the pencil is
touching 2 points, the graph did not pass the test.
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Pre Calculus Chapter 1
Example 3: Use the vertical line test to determine whether y is a function of x.
a.
b.
c.
Example 4: Select the best viewing window that shows the most complete graph of the function 𝑓(𝑥) =
−0.2𝑥 2 + 3𝑥 + 32
A. [-2, 20] by [-10, 30]
B. [-10, 30] by [-5, 50]
C. [0, 10] by [0, 200]
Increasing and Decreasing Functions
By looking at the graph to the left from left to right, the function is said to
be decreasing on the interval (-2, 0). The function is constant on the
interval (0, 2), and the function is increasing on the interval (2, 4).
REMEMBER TO READ THE GRAPH FROM LEFT TO RIGHT!
Relative Minimums and Maximums
The points at which a function changes it’s behavior (increasing,
decreasing or constant) is helpful in finding the minimum and maximum
values of a function.
By looking at the function to the right, you can see that there is a low point and a high point on the graph.
The high point is called a relative (local) maximum.
The low point is called a relative (local) minimum.
When asked for the minimum or maximum value of a function,
you are asked for the y-value.
Example 5: Find all relative maximum and minimum of the function and all intervals of increasing and
decreasing.
𝑔(𝑥) = 2𝑥 3 + 3𝑥 2 − 12𝑥
Relative Maximum: ___________________________
Relative Minimum: ____________________________
Increasing: __________________________________
Decreasing: _________________________________
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Pre Calculus Chapter 1
Example 6: Sketch the piecewise function by hand, and then find its
relative maximum and relative minimum and all intervals of
increasing and decreasing.
−𝑥 2 + 1,
𝑥≤1
𝑓(𝑥) = �
𝑥>1
√𝑥 − 1,
Relative Maximum: ___________________________
Relative Minimum: ____________________________
Increasing: __________________________________
Decreasing: _________________________________
Even and Odd functions
A function is even if it has symmetry with respect to the y-axis.
A function is odd if it has symmetry with respect to the origin. (y = x)
http://demonstrations.wolfram.com/EvenAndOddFunctions/
TO TEST IF EVEN OR ODD:
Even Function: For each x in the domain of f, 𝑓(−𝑥) = 𝑓(𝑥)
Odd Function: For each x in the domain of f, 𝑓(−𝑥) = −𝑓(𝑥)
If testing using x is confusing try substituting an actual value in for x. Try a number and its opposite. For
instance, try 2 and -2 or try 5 and -5. If you get the same results with the number and its opposite then the
function is even. If you get opposite results, the function is odd. If your results are not the same or opposite,
then the function is neither (even nor odd).
Example 7: Determine if the following functions are even, odd, or neither using a calculator.
a.
5𝑥
𝑥 2 +1
b. √1 − 𝑥
Example 8: Determine if the following functions are even, odd, or neither, using algebraic methods.
a.
1
3
𝑥 6 − 2𝑥 2
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Pre Calculus Chapter 1
b. 𝑥 3 − 5𝑥
Greatest Integer Function: denoted by ⟦𝑥⟧, defined as the greatest
integer less than or equal to x.
y =[x]
Integer means __________________________________
⟦−1⟧
= greatest integer less than or equal to -1 =
1
�10�=
⟦1.5⟧ =
⟦−1.5⟧=
Review Topic:
Distance Formula: 𝑑 = �(𝑦2 − 𝑦1 )2 + (𝑥2 − 𝑥1 )2
Midpoint Formula: �
𝑥1 +𝑥2 𝑦1 +𝑦2
2
,
2
�
Find the distance and the midpoint between the two points: (3, -7), (-4, 5)
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Pre Calculus Chapter 1
Extra Examples:
1. Sketch a rough example of a function that is increasing on the interval (-∞, 0), constant on the interval
(0,2) and decreasing on the interval (2, ∞)
2. Use a graphing utility to graph the function and determine the open interval on which the function is
increasing, decreasing, or constant.
𝑓(𝑥) = 𝑥√𝑥 + 3
3. Use a graphing utility to approximate any relative minimum or relative maximum values of the
function.
𝑓(𝑥) = 𝑥 2 − 6𝑥
4. Algebraically determine whether the function is even, odd, or neither. Verify your answer using a
graphing utility.
𝑓(𝑡) = 𝑡 2 + 2𝑡 − 3
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Pre Calculus Chapter 1
1.4 Shifting, Reflecting, and Stretching Graphs (Transformations)
In this section, you will learn how to make Transformations to basic graphs.
Transformations
Making adjustments to basic graphs is a critical skill in PreCalculus. Knowing how to adjust graphs from the
basic (parent) graph will save you a lot of time. You should learn how to adjust a basic graph of 𝑦 = 𝑓(𝑥) in
the following ways.
Vertical and Horizontal Shifts
• OUTSIDE the grouping symbols moves the graph up or down
o 𝑦 = 𝑓(𝑥) + 𝑑
 f(x) moves upward if 𝑑 > 0 and downward if 𝑑 < 0
•
INSIDE the grouping symbols moves the graph left or right in the opposite direction than the sign
suggests
o 𝑦 = 𝑓(𝑥 + 𝑐)
 f(x) moves right if 𝑐 < 0 and left if 𝑐 > 0
Reflections
• NEGATIVE leading integer flips the graph
o 𝑦 = −𝑓(𝑥) flips or reflects the graph across the x-axis
o 𝑦 = 𝑓(−𝑥) flips or reflects the graph across the y-axis
Stretch and Squeeze (a.k.a. Skinny and Fat)
• Leading coefficients cause stretches or squeezes
o 𝑦 = 𝑎 ∙ 𝑓(𝑥)
 Stretch (|𝑎| > 1) or Squeeze (|𝑎| < 1) the graph of 𝑦 = 𝑓(𝑥) vertically from the x-axis.
Points on the x-axis stay where they are. If 𝑎 < 0 (negative) then the graph must also
be reflected across the x-axis.
o 𝑦 = 𝑓(𝑏 ∙ 𝑥)
 Stretch (|b|<1) or Squeeze (|b|>1) the graph of y=f(x) horizontally from the y-axis
(multiplies x-values by the reciprocal). Points on the y-axis stay where they are. If b<0
(negative), then the graph must also be reflected across the y-axis.
**ANYTHING inside Parenthesis (or brackets, or roots) affects the function HORIZONTALLY (x-values) AND
has the OPPOSITE EFFECT!!!
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Pre Calculus Chapter 1
Absolute Value Transformations
• An absolute value can produce reflections on portions of the parent function graph
o y=|f(x)|
 Reflect the portion of the graph y=f(x) which is below the x-axis across the x-axis. Leave
the portion above the x-axis alone. (The graph will contain no negative y-values)
o y=f(|x|)
 Eliminate completely the portion of the graph y=f(x) which is to the left of the y-axis. In
its place, draw the mirror image (with respect to the y-axis) of the portion of the graph
to the right of the y-axis.
(When finished, your graph should always be symmetric to the y-axis with the side to
the right of the y-axis unchanged)
**WARNING** Graphing 𝑦 = |𝑎 ∙ 𝑓(𝑏𝑥 + 𝑐) − 𝑑| or some other combinations of the above adjustments may
be hazardous to your mental health.
None the less, we should be able to do at least some simple combinations of the above adjustments. Good
Luck!
Parent Graph: The original function before any shifts. (See page 42 for some typical parent graphs.)
The following are some of the basic graphs you will be working with throughout the year.
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Pre Calculus Chapter 1
Example 1: Compare the graph of each function with its parent graph. Sketch the parent graph and then
sketch the transformed graph
A. 𝑦 = |𝑥| − 4
C. 𝑓(𝑥) = (𝑥 + 2)3
B. 𝑦 = (𝑥 − 5)2 + 2
Parent Graph: ___________
Parent Graph: ___________
Parent Graph: ___________
Transformed Graph:
22
Pre Calculus Chapter 1
Transformed Graph:
Transformed Graph:
a
Example 2:
Use the graph of f to sketch each graph.
b
c
e
f
23
Pre Calculus Chapter 1
d
g
Example 3: Find an equation for each graph shown below and describe the transformation(s). The parent
function is 𝑦 = 𝑥 2
Transformation(s):________________________ Transformation(s):________________________
______________________________________
______________________________________
Equation: ______________________________
Equation: ______________________________
Example 4: Compare the graph of the function with the graph of 𝑦 = √𝑥.
a. 𝑦 = √𝑥 + 2
b.
𝑦 = √−𝑥 + 3
c.
𝑦 = 3 − 2√𝑥 − 3
Review Topic:
1. Determine whether the lines L1 and L2 are parallel, perpendicular or neither.
L1: (.2, .2) (2, 10)
L2: (-1, 3) (3, 9)
2. Find the domain of the function.
𝑓(𝑥) = �100 − 𝑥 2
24
Pre Calculus Chapter 1
Extra Examples:
1. Compare these equations graphically on a graphing calculator:
1
𝑦 = 𝑥 2 , 𝑦 = 3𝑥 2 and 𝑦 = 3 𝑥 2
2. Sketch the graph of the three functions by hand on the same rectangular coordinate system. Verify
your result with a graphing utility
•
•
•
𝑓(𝑥) = −𝑥 2
𝑔(𝑥) = −𝑥 2 + 1
ℎ(𝑥) = −(𝑥 − 2)2
3. Identify the parent function and describe the transformation shown. Write an equation for the
graphed function.
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Pre Calculus Chapter 1
1.5 Combinations of Functions
In this section, you will learn to combine and compose two functions together and find the domain of the
result.
Combining Functions
Definitions of Sum, Difference, Product and Quotient of Functions (Combinations)
Let f and g be two functions with overlapping domains. Then, for all x common to both
domains the Sum, Difference, Product and Quotient of f and g are defined as follows:
Sum (Adding functions)
Difference (Subtracting functions)
Product (Multiplying functions)
Quotient (Dividing functions)
(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
(𝑓 – 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
(𝑓𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥)
(𝑓/𝑔)(𝑥) = 𝑓(𝑥)/𝑔(𝑥)
𝑔(𝑥) ≠ 0
Example 1: Given 𝑓(𝑥) = 2𝑥 + 1 and 𝑔(𝑥) = 𝑥 2 + 2𝑥 − 1, find the following:
A. 𝑓(𝑥) + 𝑔(𝑥)
(𝑓 + 𝑔)(1)
B. 𝑓(𝑥) − 𝑔(𝑥)
(𝑓 − 𝑔)(0)
C. 𝑓(𝑥)𝑔(𝑥)
(𝑓𝑔)(−1)
D. 𝑓(𝑥)/𝑔(𝑥)
(𝑓/𝑔)(2)
Example 2: If f(x) = x2 and g(x) = 1 - x, find:
(𝑔 − 𝑓)(𝑥)
(𝑓 − 𝑔)(𝑥)
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Pre Calculus Chapter 1
(𝑔/𝑓)(𝑥) What is the domain of g/f?
(𝑓/𝑔)(𝑥) What is the domain of f/g?
Composing Functions
The composition of the functions f and g is
(𝑓 ∘ 𝑔)(𝑥) = 𝑓�𝑔(𝑥)�
The domain of 𝑓 ∘ 𝑔 is the set of all x in the domain of g such that g(x) is the
domain of f.
(x must be in the domain of g, and g(x) must be in the domain of f.)
Composite Function: Putting one ______________ into another.
Example 3: Find 𝑓 ∘ 𝑔, 𝑔 ∘ 𝑓 and (𝑓 ∘ 𝑔)(0), if 𝑓(𝑥) = 3𝑥 + 5 and 𝑔(𝑥) = 5 – 𝑥.
a. 𝑓 ∘ 𝑔 =
b. 𝑔 ∘ 𝑓 =
c. (𝑓 ∘ 𝑔)(0) =
Example 4: Let 𝑓(𝑥) = 𝑥 2 − 1 and 𝑔(𝑥) = √𝑥. Find the following:
a. 𝑓�𝑔(𝑥)� =
Domain: ______________
b. 𝑔�𝑓(𝑥)� =
Domain: ______________
c. 𝑓�𝑔(9)� =
d. 𝑔�𝑓(3)� =
27
Pre Calculus Chapter 1
Example 5: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius
(in feet) of the outer ripple is given by 𝑟(𝑡) = .6𝑡, where 𝑡 is the time in seconds after the pebble strikes the
water. The area of the circle is given by the function 𝐴(𝑟) = 𝜋𝑟 2.
A. Find and interpret (𝐴 ∘ 𝑟)(𝑡)
B. Use your calculator to graph the area as a function of 𝑡. Find the time required for the area enclosed
by a ripple to increase to 20 ft2 (Hint: draw a horizontal line at 20 ft2 and find where the two intersect
using the intersect feature on your calculator.)
Review Topic:
1. Find an equation of the line that passes through the two points.
(-4, 2) (3, 8)
2. Find 3 points that lie on the graph of the equation: 𝑦 = | − 𝑥 2 − 3𝑥 + 2|
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Pre Calculus Chapter 1
Extra Examples:
1. Use the graphs of 𝑓 and 𝑔 to graph ℎ(𝑥) = (𝑓 + 𝑔)(𝑥).
2. Evaluate the indicated function for 𝑓(𝑥) = 𝑥 2 − 1 and 𝑔(𝑥) = 𝑥 − 2 algebraically. If possible, use a
graphing calculator to verify your answer.
(𝑓 + 𝑔)(𝑡 − 4) =
(𝑓𝑔)(4)=
3. Determine the domains of 𝑓, 𝑔 and 𝑓 ∘ 𝑔. Use a graphing utility to verify your results.
𝑓(𝑥) = √𝑥 + 3, 𝑔(𝑥) =
𝑓(𝑥) =
𝑥
2
2
, 𝑔(𝑥) = 𝑥 − 1
|𝑥|
𝑓(𝑥) = 𝑥 + 3
29
𝑔(𝑥) = 𝑥 2 + 𝑥
Pre Calculus Chapter 1
1.6 Inverse Functions
In this section, you will learn to find inverses of functions, graph functions and their inverses and determine if
an inverse is a function. You will also learn what a one-to-one function is.
One-to-One Function
A function f is one-to-one if and only if every horizontal line intersects the
graph of f in at most one point.
Example 1: Determine which functions below are one-to-one.
A. 𝑓(𝑥) = 𝑥 2 + 4
B. 𝑓(𝑥) = 𝑥 3
C. 𝑓(𝑥) = √𝑥 − 2
HINT: 1. A function that is increasing throughout its domain is one-to-one.
2. A function that is decreasing throughout its domain is one-to-one.
Definition of the Inverse of a Function
Let f and g be two functions such that
1. 𝑓(𝑔(𝑥)) = 𝑥
2. 𝑔(𝑓(𝑥)) = 𝑥
Then, functions f and g are inverse of each other.
** The inverse function is denoted by 𝑓 −1 or 𝑓 −1 (𝑥)
The domain of f must be equal to the________ of f-1. And
The range of f must be equal to the _________of f-1
Finding the Inverse of a Function
To find the inverse of a function use the following steps:
1. Replace 𝑓(𝑥) with 𝑦
2. Interchange 𝑥 and 𝑦
3. Solve the new equation for 𝑦  This new equation is 𝑓 −1 (𝑥)!
If the new equation cannot be solved for 𝑦 explicitly, then 𝑓(𝑥) does not have an inverse
function.
ALL FUNCTIONS HAVE AN INVERSE; however, not all inverses are functions! If a function is one-to-one, then
that function also has an inverse which is a function.
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Pre Calculus Chapter 1
3
Example 2: Given 𝑓(𝑥) = 2 − 2 𝑥 answer the following:
A. Is 𝑓(𝑥) one-to-one? ______
B. Find the inverse of 𝑓(𝑥)
C. Verify that the equation you got above for the inverse is indeed the inverse.
D. Graph 𝑓(𝑥), 𝑓 −1 (𝑥) and 𝑦 = 𝑥 on the same graph.
(use different colors for each function if possible)
E. Pick at least 2 points (𝑥, 𝑦) on the graph of 𝑓 and verify
that (𝑦, 𝑥) is on the graph of 𝑓 −1 (𝑥).
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Pre Calculus Chapter 1
Inverse Functions: If a function f is one-to-one, then its inverse 𝑓 −1 (𝑥) is a function. The graph of 𝑓 contains
the point (𝑥, 𝑦) if and only if the graph of 𝑓 −1 (𝑥) contains the point (𝑦, 𝑥). Thus the graph of 𝑓 −1 (𝑥) is a
reflection of the graph of 𝑓 across the line 𝑦 = 𝑥
Example 3: Show that f and g are inverse functions algebraically.
3
𝑓(𝑥) = 2𝑥 3 − 1 𝑎𝑛𝑑 𝑔(𝑥) = �
𝑥+1
2
Now use a graphing utility to create a table of values for each function to numerically show that f and g are
inverses.
Review Topic:
1. Write the rational expression in simplest form.
𝑥 2 − 16
4−𝑥
2. Determine whether the equation represents y as a function of x.
𝑥2 𝑦2
+
=1
4
9
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Pre Calculus Chapter 1
Extra Examples:
1. Show that 𝑓 and 𝑔 are inverses algebraically. Use a graphing utility to graph 𝑓 and 𝑔 in the same
viewing window. Describe the relationship between the graphs.
1
1
𝑓(𝑥) = 𝑎𝑛𝑑 𝑔(𝑥) =
𝑥
𝑥
2. Find the inverse function of 𝑓 algebraically. Use a graphing utility to graph both 𝑓 and 𝑓 −1 in the same
viewing window. Describe the relationship between the graphs.
𝑓(𝑥) = 𝑥 3 − 4
3. Restrict the domain of the function 𝑓 so that the function is one-to-one and has an inverse function.
Then find the inverse function 𝑓 −1 . State the domains and ranges of 𝑓 and 𝑓 −1 . Explain your results.
(There are many correct answers)
𝑓(𝑥) = (𝑥 + 3)2
33
Pre Calculus Chapter 1
1.7 Linear Models and Scatter Plots
Fitting a Line to Data means finding a linear model or equation to represent the relationship described by a
scatter plot.
Interpreting
Correlation:
Example 1: Determine whether there is positive correlation, negative correlation or no discernible correlation
between the variable.
Fitting a line to Data gives you the Line of Best Fit.
Year
1998
1999
2000
2001
2002
2003
2004
Household credit market
debt, D (in trillions of dollars)
6.0
6.4
7.0
7.6
8.4
9.2
10.3
Example 2: Find a linear model that relates the year, t, with the amount of outstanding credit debt, D. Let t=8
correspond to the year 1998.
34
Pre Calculus Chapter 1
Steps (for TI-84):
Use your calculator to form the scatter plot
STAT  EDIT  List your values in L1 and L2
**Make sure your stat Plot is on!
Use the Linear Regression feature to find the model or the line of best fit
STAT  CALC  LinReg (ax+b)  Enter  Enter
Now you are given a formula and the values to plug into the formula for a and b.
To paste the equation into Y1= add a couple steps
STAT  CALC  LinReg (ax+b)  L1, L2  varsy-varsfunctionY1
Correlation Coefficient:
This is the values r that the calculator gives. Correlation Coefficients vary between -1 and 1, the
closer |r| is to 1, the better the points can be described by a line.
(make sure your diagnostics is on: catalog D  Diagnostic On  Enter Enter)
How does the correlation coefficient in example 2 describe our data?
Example 3: Find a linear model that relates the year t with the number of people (in millions) P in the United
States labor force. Let t = 3 be the year 1983.
t
3
4
5
6
7
8
9
10
11
12
13
P
113
115
117
120
122
123
126
126
127
129 130
a. Create a scatter Plot
b. Find the linear regression model
c. How closely does the model fit the data?
d. Use the model to predict the population in the US Labor force for the year 2000
e. Use the model to predict the population in the US Labor force for the year 2010
Review
Solve the equation algebraically. Check your solution graphically.
10𝑥 2 − 23𝑥 − 5 = 0
35
Pre Calculus Chapter 1