2.1 – Linear Equations
In Algebra 1 and Geometry we were taught to display linear equations predominately in slope-intercept form, y = mx + b .
Although slope-intercept form will still be used this year, three other forms of displaying equations of lines is required in A2PCH
and Calculus.
Four Forms of Linear Equations
To determine the equation of a line, we need two pieces of information: a point on the line, (x1 , y 1 ) , and the slope of the line, m.
In A2PCH we will substitute both the given point and slope into the point-slope form​, y − y 1 = m (x − x1 ) , and then algebraically
manipulating the equation into the desired form of the line.
Here are the four forms that Calculus will require you to know.
1. Slope-Intercept Form
y = mx + b
2. General Form
Ax + B y + C = 0 ,
3. Point-Slope Form
y − y 1 = m(x − x1 )
4. Taylor Form
y = y 1 + m(x − x1 )
where A, B, and C are integers and A is always positive
If the directions of a math problem has stated which form you are required to display the equation of a line, expect a credit
deduction if it is not displayed properly. A2PCH is all about the details.
Determining the Slope of a Line
Before writing the equations of lines in their proper form, a quick review of the various ways to determine the slope of a line.
Given an Equation of a Line
Rewrite (if necessary) into slope-intercept form;
slope, is the coefficient of the ​x​-variable.
Given a Graph of a Line
Slope is rise over run m = ∆y
∆x .
Given Two Points on a Line
y −y
Slope is m = x2 −x1 .
Given “Parallel” or “Perpendicular”
Slopes of parallel lines are same: m1 = m2 .
3x − 2y = 4
2
1
Slopes of perpendicular lines are negative reciprocals:
m1 • m2 =− 1 or m1 =− m1 .
(2, 4) and (4, − 4)
2
4x − y = 3
1
Example 1
Determine the equation of a line that passes through the points (− 5, 1) and (3, − 7). Display the answer in (a)
Taylor form, (b) slope-intercept form, and (c) general form.
Example 2
Determine the equation of a line that passes through the point (4, -3) and is (a) parallel and (b) perpendicular to the
line 5x − 2y = 3. Then find the (c) ​x​-intercept of the perpendicular line.
Example 3
True or False? The lines represented by ax + by = c1 and bx − ay = c2 are perpendicular. Assume a≠0 and
b≠0. Justify your answer.
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2.2 – Functions
A linear equation, like y = 3x + 2, shows the relationship between an input number (represented by the x variable) and the
output number (represented by the y variable).
Input
Relationship
Output
−4
× 3 then + 2
− 10
−2
× 3 then + 2
−4
0
× 3 then + 2
2
1
× 3 then + 2
5
3
× 3 then + 2
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We can show this relationship between the input and output by displaying them as an ordered pair such as (− 4, − 10) or (0, 2) .
A point is called an ordered pair because the input always comes first and the output second: ( input , output ) .
Definition of a Function
A function has a special relationship between the input and output.
● The relationship must work for every possible input value (the x -coordinate).
● For each input value (the x -coordinate), there is ​exactly​ one output value (the y -coordinate).
Graphical Testing for Functions: The Vertical Line Test
When given the sketch of a graph, see if the graph passes the Vertical Line Test (VLT).
Numerical Testing for Functions: Mapping
When given a table of values, check to see that for each x -value there is ​exactly one y -value OR graph out the points and see if
it passes the VLT.
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Analytical Testing for Functions: Isolate the y -Coordinate
When given an equation, isolate the y -coordinate. If y = ±√(x expression) , then the equation is not a function (since for each x
-value there are two y -values); if y = (x expression) , then the equation is a function (since for each x -value there is exactly one
y -value).
x + 2y = 6
x2 + y 2 = 16
Testing Summary
● Graphically, we see if it passes the vertical line test.
● Numerically, we see that for each value of x , there is only one value of y .
● Analytically, we isolate the y -variable to see that for each value of x , there is only one value of y .
Function Notation f (x)
If an equation is also a function, we mathematicians want to know that it’s a function without having to perform any of the tests
previously discussed. To inform the reader that an equation is a function, we replacing the variable y with the symbol f (x) . The
notation f (x) is read as “ f of x ” where the symbol f stands for function. Here are some examples.
y = x2
⟶
f (x) = x2
y = 2x + 1 ⟶
f (x) = 2x + 1
y = √x − 4 ⟶
f (x) = √x − 4
Do not read​ the parentheses in f (x) as a form of multiplication! It is a place holder for the input. In the function f (x) = x2 ,
what goes into the function is put inside the parentheses after the name of the function on the left side of the equal sign while
what the function does with the input (i.e. the output) is shown on the right side of the equal sign.
An input of x = 4 becomes an output of 16 :
f (4) = 42
f (4) = 16
Since f (x) = y , the output is the y -value corresponding to the x -value of the input – an ordered pair (x, y) or (x, f (x)) . In this
case (4, 16) .
● The input is x .
● The relationship is to square each input.
● The output is y .
yg
4
To illustrate how the substitution of f (x) for y helps us in math, suppose we are interested in finding the ​y-​ value of the function
y = 2x + 1 when x = 3 . Below is a comparison of this evaluation using y and f (x) .
A.
Find the y -value of the function y = 2x + 1 when x = 3 .
B. Find f (3) given f (x) = 2x + 1
It is convenient to name a function so that it can be referenced easily. The name can be chosen from any of our English letters,
both upper and lower cases, and is always in ​italic​. The most common name for a function is f (x) ; other frequently used names
are g (x) and h(x).
Example 1
A.
f (2)
Example 2
A.
g (1)
If f (x) =− 3x + 1 , evaluate each of the following. Then interpret them geometrically.
B. f (− 4)
C. f (0)
If g (x) = x2 + 2x + 1 , evaluate each of the following.
B. g (2x)
C. g (4x)
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The Domain of a Function
The set of all possible x -values (i.e. all the possible inputs) for a function is called its domain.
Although sketching a graph of a function allows us to determine its domain, it isn’t very efficient. When given an equation of a
function, it is best to determine its domain algebraically by looking for x -values that ​don’t work​, then use this information to
find the x -values that work.
In general, the domain of a function ​excludes x -values​ that would cause
● a denominator to equal 0
● a radical of even index to equal a negative number.
Example 5
A.
Find the domain of each function. Display the domain in interval notation.
f (x) =− 32 x + 1
B. g (x) = √x − 2
3x
C. f (x) = x+4
2x
D. h (x) = √x+3
F. f (x) = √2x−4
4
√
3
E. g (x) = √x − 5
x−1
So Many Names!
Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have
come about. Here are some common terms with which to get familiar:
Input
x, t, …
Relationship
f , g, h, …
Domain
Independent Variable
Argument
Output
f (x) , g (t) , h (x) , …
Range
Dependent Variable
Value of a Function
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2.3 – Naming Important Characteristics of Graphs
When sketching a graph of a function or studying a graph of a function, we need words that describe its important characteristics.
Increasing and Decreasing
When we “read” a graph for its important characteristics, we must read it like a sentence: from left to right. As we read a graph
from left to right we are always looking at increasing x -values. Increasing and decreasing describe what is happening to the y
-values​ as we read the graph from left to right.
● Decreasing
● Increasing
● Constant
y -values are decreasing when reading the graph from left to right (the graph goes down-hill)
y -values are increasing when reading the graph from left to right (the graph goes up-hill)
y -values are neither increasing nor decreasing when reading the graph from left to right (the graph is flat)
Although we describe where a function is increasing or decreasing based upon the change in the y -values as the graph moves
from left to right, increasing and decreasing intervals are displayed in terms of the x -values.
Example 1
Use the graphs to determine the intervals over which the function is increasing, decreasing, or constant. Display
the answers in interval notation.
Should the endpoints for the graph on the right be included in the increasing or decreasing intervals? To avoid the debate over
whether the endpoints should or should not be included (the answer is dependent on knowing calculus), we will not include them
this year. So all increasing and decreasing intervals will use parentheses (x1 , x2 ) , not brackets [x1 , x2 ] .
Constant intervals will be taken up in the next lecture while studying piece-wise functions.
Domain and Range
The ​domain​ of a function is the set of all possible x -values of its graph (“input” values).
The ​range​ of a function is set of all possible y -values of its graph (“output” values).
Example 2
Find the domain and range of the two graphs shown in Example 1.
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Zeros of a Function
If the graph of a function has an x -intercept at (a, 0) , then a is called the zero ​of the function since the x -value of a yields a y
-value of 0. To find the zeros of a function, set f (x) = 0 , and solve for all values of x.
Example 3
Find the zeros of the following functions.
A.
g (x) = 4x − 2
B. F (x) = 3x2 + x − 10
C.
f (x) = √16−x
x
D. H (x) = x3 − 4x2 − 9x + 36
2
Feeling a bit more comfortable using f (x) instead of y ? Here is a popular calculus problem that you are now able to solve.
Example 4
Write the linear function that has the function values of f (3) = 9 and f (− 1) =− 11 leaving your answer in
general form.
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2.3 – More on Naming Important Characteristics of Graphs
Odd and Even Functions
In chapter 1 we studied three different types of symmetry for a graph:
x -axis symmetry
y -axis symmetry
Origin symmetry
A graph of an equation is symmetric with respect to the x -axis if replacing y with –y yields the same
equation.
A graph of an equation is symmetric with respect to the y -axis if replacing x with –x yields the same
equation.
A graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields
the same equation.
A function cannot have x -axis symmetry since by definition it would not pass the Vertical Line Test. When talking about
functions, we generally use different names to describe y -axis and origin symmetry: even function and odd function.
Exhibits ​y​-axis symmetry
Exhibits origin symmetry
Even function
Odd function
To determine algebraically whether a function is even or odd without sketching a graph, use the following function notation tests.
● A function f (x) is ​even​ if f (− x) = f (x) . When we input − x into the function f , we get the same function back.
● A function f (x) is ​odd if f (− x) =− f (x) . When we input − x into the function f , we get a negative of the same function
back.
Example 1
Determine algebraically if the following functions are even, odd, or neither.
A.
f (x) = x2 − x4
B. h (x) = 3x3 − x
C.
g (x) = x2 − x4 + 1
D. F (x) = 3x3 − x − 2
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Determining When f (x)≥0 Without Graphing
When we are asked to determine the interval for which f (x)≥0 , it is easiest to find it algebraically, without graphing out the
function.
Example 2
Determine the interval for which f (x)≥0 given​ f (x) = 4x + 3 (a) graphically and (b) algebraically.
(a)
Example 3
(b)
Determine the interval for which f (x)≥0 algebraically for each of the following functions.
(B) g (x) = √x + 2 − 3
A. f (x) = x2 − 3x
Tonight’s assignment will ask you to graph out these function to see where f (x)≥0 . Override this instruction and do them
algebraically, if possible.
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2.4 – Piecewise Functions
A piecewise function behaves differently based on the input value ( x -values). Their “pieces” may be all linear or a combination
of equation forms (such as constant, linear, quadratic, square root, etc…). Here are some graphs of piecewise functions.
To sketch a graph of a piecewise function, we need to graph of each “piece” for its given set of x -values.
Example 1
Graph the following piecewise functions. Then determine its domain, range, where it is increasing, and where it is
decreasing.
A.
f (x) = {2x + 1, x≤ − 1; x2 − 3, x >− 1
B.
g (x) = {√x + 2, x < 0; x + 1,
x≥0
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The Absolute Value Function
The absolute value function is a famous piecewise function. It has two pieces: one piece below x = 0 and another from x = 0
and above.
Example 2
Sketch a graph the function f (x) = |x| = {x,
x≥0; − x, x < 0 .
The Step Function
Another famous piecewise function is the step function. We can see where the function got its name by looking at the graph of
the step function, f (x) = ⟦x⟧ .
What is this ⟦⟧ notation? The notation is called the greatest integer finder.
● If a number inside the notation is an
●
integer,
use that integer:
⟦2⟧ = 2, ⟦10⟧ = 10, ⟦ − 5⟧ =− 5, ⟦0⟧ = 0
If a number inside the notation is not an integer, use the next smaller integer: ⟦3.49⟧ = 3, ⟦ − 2.5⟧ =− 3, ⟦0.197⟧ = 0
To graph a step function, find the pattern of two or three steps using an x | y table, then repeat the pattern to fill in the coordinate
plane.
Example 2
Graph the function g (x) = ⟦x − 2⟧ + 1
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2.5 – Transformation of Parent Graphs
In chapter 1 we learned that it was easier to sketch a graph using a known formula (like y = mx + b for lines) than by point
plotting (using an x | y table and supplementing with intercepts and symmetry). Using a formula allows us to sketch a graph
quickly and accurately. We like that! Transformation of parent graphs is another method that allows us to sketch a graph quickly
and accurately. It essentially works like this: if we know what the graph of y = √x looks like, and we understand transformations,
we will quickly know what the graphs of y = √x + 3 and y = 2√x − 1 look like.
The first thing you need to sketch graphs using transformations is to memorize the shapes and key points of the six most
commonly used functions in algebra. These functions are collectively called ​parent functions​.
(− 2, − 2)
(0, 0)
(2, 2)
(− 4, 4)
(0, 0)
(4, 4)
(− 2, 4)
(− 1, 1)
(0, 0)
(1, 1)
(2, 4)
Vertical Shift
Suppose that f (x) is a parent function and that d > 0.
● The graph of y = f (x) + d is the graph of y = f (x) shifted d units upward.
● The graph of y = f (x) − d is the graph of y = f (x) shifted d units downward.
Note:
The transformation affects the y -coordinate only​.
The graph moves up or down the ​way you expect it to do so.
Horizontal Shift
Suppose that f (x) is a parent function and that c > 0.
● The graph of y = f (x + c) is the graph of y = f (x) shifted c units to the left.
● The graph of y = f (x − c) is the graph of y = f (x) shifted c units to the right.
Note:
The transformation affects the x -coordinate only​.
The graph moves to the left or right, the ​opposite of the way you expect it to do so.​
Example 1
A.
Sketch the graph of each function using transformations.
h (x) = (x − 1)2 + 2
B. h (x) = |x + 1| − 3
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(− 2, − 8)
(− 1, − 1)
(0, 0)
(1, 1)
(2, 8)
(0, 0)
(1, 1)
(4, 2)
(9, 3)
Vertical Reflection
Suppose that f (x) is a parent function.
● The graph of y =− f (x) is the graph of y = f (x) reflected across the x -axis.
Note:
The transformation affects the y -coordinate only​.
Each y -coordinate is multiplied by − 1 , just the ​way you expect it to do so​.
Horizontal Reflection
Suppose that f (x) is a parent function.
● The graph of y = f (− x) is the graph of y = f (x) reflected across the y -axis.
Note:
The transformation affects the x -coordinate only​.
Each x -coordinate is divided by − 1 , the ​opposite way of what you expect it to do so
Example 2
A.
Sketch the graph of each function using transformations.
f (x) =− √x
B. G (x) = √− x
Vertical Stretch/Compression
Suppose that f (x) is a parent function.
● The graph of y = a·f (x) is the graph of y = f (x) where each y -coordinate of f (x) has been ​multiplied​ by a.
Note:
The transformation affects the y -coordinate only​.
Each y -coordinate is multiplied by a , just the ​way you expect it to do so​.
Horizontal Stretch/Compression
Suppose that f (x) is a parent function.
●
The graph of y = f (bx) is the graph of y = f (x) where each x -coordinate of f (x) has been ​divided​ by b.
Note:
The transformation affects the x -coordinate only​.
Each x -coordinate is divided by b , the ​opposite way of what you expect it to do so
Example 3
Sketch the graph of each function using transformations.
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A.
h (x) = 2x2
B. g (x) = |3x|
Summary of Transformations
How then do we graph a function that exhibits all three types of transformations – shifts, reflections, stretches, and compressions?
Given the transformation function, y = a·f ( b (x + c) ) + d , where f (x) is the parent function.
Vertical Transformations
● a is a vertical stretch/compression
o
a>1
Stretches the graph in the y -direction (multiply each y -coordinate by a ).
●
o
0<a<1
Compresses the graph in the y -direction (multiply each y -coordinate by a ).
o
− 1 < a < 0 Reflects the graph about the x -axis
& compresses the graph in the y -direction (multiply each y -coordinate by − a ).
o
a <− 1
Reflects the graph about the x -axis
& stretches the graph in the y -direction (multiply each y -coordinate by − a ).
d is a vertical shift
o
d>0
o
d<0
Shifts the graph upward (add d to each y -coordinate).
Shifts the graph downward (add d to each y -coordinate).
Vertical transformations act on the y -coordinate in the way we expect them to do so.
Order of operations: Perform the a transformation before the d transformation.
Horizontal Transformations
● b is a horizontal stretch/compression
o
b>1
Compresses the graph in the x -direction (divide each x -coordinate by a ).
●
o
0<b<1
Stretches the graph in the x -direction (divide each x -coordinate by a ).
o
− 1 < b < 0 Reflects the graph about the y -axis
& stretches the graph in the x -direction (divide each x -coordinate by a )
o
b <− 1
Reflects the graph about the y -axis
& compresses the graph in the x -direction (divide each x -coordinate by − a ).
c is a horizontal shift
o
c>0
o
c<0
Shifts the graph to the left (subtract c from each x -coordinate).
Shifts the graph to the right (subtract c from each x -coordinate).
Horizontal transformations act on the x -coordinate in the opposite way we expect them to do so.
Order of operations: Perform the b transformation before the c transformation.
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Example 4
Sketch the graph of each function using transformations.
A.
f (x) =− (x − 1)2 + 2
B. g (x) = √x + 4 − 2
C.
h (x) = (x + 1)3 − 3
D. f (x) = 21 |3x − 3| + 1
Example 5
When A2PCH and Calculus teachers want to check for understanding of transformations, they will often give
you a piecewise function to transform.
Original function f (x)
A. g (x) = f (x − 1) + 3
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B. g (x) =− 2f (x) + 1
2.6 – Building Functions from Other Functions
Just as we can add, subtract, multiple, and divide two real numbers together to get a new number, two functions can be combined
to create a new function.
Sum, Difference, Product, and Quotient of Functions
● Sum:
f (x) + g (x) = (f + g ) (x)
·​ Product:
f (x) ·g (x) = (f g) (x)
● Difference:
f (x) − g (x) = (f − g )(x)
·​ Quotient:
f (x)
g(x) =
( )x
f
g
The domain of the resulting function is the x -values common to both domains.
Example 1
A.
Let f (x) = 2x − 1 and g (x) = x2 − 2 . Determine the resulting function and their domain.
(f + g ) (x)
Example 3
B. (f g) (x)
Let g (x) = √x + 3 and h (x) = √x − 2 . Determine the resulting function g + h and its domain.
Function Notation Review
Another way of combining two functions is by taking their composite. To help understand this new combination, let us review
function notation from section 2.2.
Example 4
Use f (x) = 3x − 4 to find the following function values.
A.
f (3)
B. f (− 2)
C.
f (4x)
D. f (x + 1)
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Composition Functions
The x in the notation f (x) is a placeholder for various inputs into a relationship defined by f (x) . Using the function from
example 4,
f (x) = 3x − 4
f (input) = 3(input) − 4
In a composition function, the input is another function. The mathematical notation indicating a composition is ∘ .
The Composition of Functions
(f ∘g) (x) = f ( g (x) )
The domain of the resulting function is the x -values common to both the domain of the input function and the domain of the
output function.
Example 5
Let f (x) = x2 − 1 and g (x) = x + 4 . Find the each of the following functions and its domain.
A.
(f ∘g) (x)
B. (g∘f ) (x)
C.
(f ∘f ) (x)
D. (g∘g) (2)
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Determining the Domain of a Composition Function
In the previous example, both of the given functions had the domain of all real numbers, ( − ∞, ∞) , so the resulting composition
function had the same domain of all real numbers, ( − ∞, ∞) .
When one or more of the given functions has a domain restriction, however, we need to
● find the domain of the input function
● find the domain of the output function
● the x -values common to the domain of the input function and to the domain of the output function is the domain of the
resulting composite function.
Example 6
A.
Let f (x) = 1x and g (x) = √x + 2 . Determine the domain of the following functions.
f ∘g
B. g ∘f
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2.7 – The Inverse of a Function
An inverse function goes the other way of the relationship defined by f (x) .
If we have the function f (x) = 2x + 3 , the relationship of the input x is
The inverse function goes the other
way:
The mathematical notation for inverse is a little “ − 1 ” after the function name like f −1 (y) . We say “ f inverse of y . ” So the
inverse of f (x) = 2x + 3 is f −1 (y) = 21 y − 23 .
The placement of - 1 to the “exponent position” has two different meanings depending on what the “base” is.
● If the “base” is a number or variable, the - 1 indicates a reciprocal operation such as 2−1 = 21 or x−1 = 1x .
● If the “base” is a function, the - 1 indicates a inverse of a function as shown above. [The reciprocal of f (x) is f (x)−1 ]
Back to Where We Started
An inverse f (x) = 2x + 3 is f −1 (y) = 21 y − 23 . . Using f (x) = 2x + 3 and f −1 (y) = 21 y − 23 from above, input x = 5.
f (5) = 2·5 + 3 = 13
We can then use the inverse on the output y = 11.
f −1 (13) = 21 ·13 − 23 = 10
2 =5
And we get 5 back again. Getting back the original value can be written in composite form:
f −1 ( f (5) ) = 5
The inverse of f (5) is 5 . Or more generally,
f −1 ( f (x) ) = x
Determining the Inverse Function
Given a function, its inverse can be determined by exchanging the x variable with its y variable, and then isolate the y variable.
f (x) = 2x + 3
y = 2x + 3
x = 2y + 3
x − 3 = 2y
x − 3 =y
2
2
y = 21 x − 23
f −1 (x) = 21 x − 23
−1
We are required to replace x with f (x) to indicate the inverse relationship. Credit deduction if missing.
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Example 1 Match each function in the first column with its inverse function in the second column.
g (x) = 3x + 2
p−1 (x) = √x − 1
h (x) = 4x − 7
g −1 (x) = 31 x − 32
p (x) = x3 + 1
5
G−1 (x) = x−4
G (x) = 5x + 4
h−1 (x) = 41 x + 47
3
The Graph of an Inverse Function
The graph of the inverse function is a reflection of the function about the line y = x .
Example 2
Find the inverse of g (x) = 3x , determine their respective domain and range, then graph them both on same
coordinate plane.
Domain Restricted of Inverse Function
An inverse function exchanges the x variable with its y variable. It is also true of domain (set of all x -values) and range (set of
all y -values). The domain of a function becomes the range of the inverse function, while the range of a function becomes the
domain of the inverse function. Domain and range exchange rolls.
Always find the domain and range of the function and exchange the roles of domain and range to determine any domain
restriction on its inverse function.
Example 3
Find the inverse of f (x) = √x − 2 , determine their respective domain and range, then graph them both on same
coordinate plane.
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An Inverse Must be a Function
Some functions do not have inverses since an inverse by definition must be function (i.e. it passes the vertical line test). To
determine if a function has an inverse, take the Horizontal Line Test (​HLT​).
Example 4
Find the inverse of g (x) = x2 + 1 , then graph them both on same coordinate plane.
Verifying Inverse Functions
To verify whether two given functions, call them f and g , are inverses of each other, we
● graph both functions to show they are reflections of each other about the line y = x (testing graphically), or
● show that ​both​ f ∘g = x and g ∘f = x (algebraic method).
Example 5
Test algebraically to determine whether f (x) = x2 and g (x) = √x are inverse functions if x≥0.
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