Chapter 1
Functions and Graphs
1.1
• Graphs & Graphing Utilities
Objectives
• Plot points in the rectangular coordinate
system
• Graph equations in the rectangular
coordinate system
• Interpret information about a graphing
utility’s viewing rectangle or table
• Use a graph to determine intercepts
• Interpret information from graphs
Rectangular coordinate system
•
•
•
•
X-axis: horizontal (right pos., left neg.)
Y-axis: vertical (up pos., down neg.)
Ordered pairs: (x,y)
Graph of an equation: infinitely many
ordered pairs that make a true statement
• X-intercept: point where y=0, (x,0)
• Y-intercept: point where x=0, (0,y)
Graphing an Equation
• List ordered pairs that make your equation
true (plug values in for x and find the
resulting y’s). Include the x-intercept & yintercept among your points.
• Plot several points and look for a trend.
• Use the graphing utility on your calculator
and compare graphs. Do they always
match? Why or why not?
Intercepts
• Intercepts are key points to plot when
graphing an equation. Remember, an
intercepts is a POINT (x,y), and not just a
number!
• Look at the x-axis: What is true of EVERY
point on the axis? (the y-value’s always 0)
• Look at the y-axis: What is true of EVERY
point on the axis? (the x-value’s always 0)
Horizontal and Vertical Lines
• What if the y-value is ALWAYS the same, regardless
of the x-value (it could be anything!). (i.e. (4,2),
(5,2), (-14,2), (17,2)) It’s a horizontal line! The x isn’t
part of the equation, because its value is irrelevant:
y=k (k is a constant)
• What if the x-value is ALWAYS the same, regardless
of the y-value (it could be anything!). (i.e. (2,4),
(2,5), (2,-14), (2,17)) It’s a vertical line! The y isn’t
part of the equation, because its value is irrelevant:
x=c (c is a constant)
1.2
• Basics of Functions & Their Graphs
Objectives
• Find the domain & range of a relation.
• Determine whether a relations is a function.
• Determine is an equation represents a
function.
• Evaluate a function.
• Graph functions by plotting points.
• Use the vertical line test to identify functions.
• Obtain information from a graph.
• Identify the domain & range from a graph.
• Identify intercepts from a graph.
Domain & Range
• A relation is a set of ordered pairs.
• Domain: first components in the relation
(independent)
• Range: second components in the relation
(dependent, the value depends on what
the domain value is)
• Functions are SPECIAL relations: A
domain element corresponds to exactly
ONE range element.
EXAMPLE
• Consider the function: eye color
• (assume all people have only one color,
and it is not changeable)
• It IS a function because when asked the
eye color of each person, there is only one
answer.
• i.e. {(Joe, brown), (Mo, blue), (Mary,
green), (Ava, brown), (Natalie, blue)}
• NOTE: the range values are not
necessarily unique.
Evaluating a function
• Common notation: f(x) = function
• Evaluate the function at various values of x,
represented as: f(a), f(b), etc.
• Example: f(x) = 3x – 7
f(2) = 3(2) – 7 = 6 – 7 = -1
f(3 – x) = 3(3 – x) – 7 = 9 – 3x – 7 = 2 – 3x
Graphing a functions
• Horizontal axis: x values
• Vertical axis: y values
• Plot points individually or use a graphing utility
(calculator or computer algebra system)
• Example:
2
y  x 1
Table of function values
y  x 1
2
X
(domain)
Y
(range)
-4
-3
-2
-1
17
10
5
2
0
1
2
1
2
5
3
10
4
17
Graphs of functions
Can you identify domain & range
from the graph?
• Look horizontally. What all x-values are
contained in the graph? That’s your
domain!
• Look vertically. What all y-values are
contained in the graph? That’s your range!
What is the domain & range of the
function with this graph?
1) Domain : (, ), Range : (, )
2) Domain : (3, ), Range : (, )
3) Domain : (3, ), Range :(3, )
4) Domain : (, ), Range : (3, )
Can you determine if a relation is a
function or NOT from the graph?
• Recall what it means to be a function: an
x-value is assigned ONLY one y-value (it
need not be unique).
• So, on the graph, if the x value is paired
with MORE than one y value there would
be two points directly on a vertical line.
• THUS, the vertical line test! If 2 or more
points fall on a vertical line that would
cross any portion of your graph, it is NOT
the graph of a function.
Finding intercepts:
• X-intercept: where the function crosses the
x-axis. What is true of every point on the
x-axis? The y-value is ALWAYS zero.
• Y-intercept: where the function crosses the
y-axis. What is true of every point on the
y-axis? The x-value is ALWAYS zero.
• Can the x-intercept and the y-intercept
ever be the same point? YES, if the
function crosses through the origin!
1.3
• More of Functions and Their Graphs
Objectives
• Find & simplify a function’s difference
quotient.
• Understand & use piecewise functions.
• Identify intervals on which a function
increases, decreases, or is constant.
• Use graphs to locate relative maxima or
minima.
• Identify even or odd functions & recognize
the symmetries.
• Graph step functions.
Difference Quotient
• Useful in discussing the rate of change of
function over a period of time
• EXTREMELY important in calculus, (h
represents the difference in two x values)
f ( x  h)  f ( x )
h
Find the difference quotient
f ( x)  2 x 3  2 x  1
f ( x  h)  2( x  h) 3  2( x  h)  1
f ( x  h)  2( x 3  3 x 2 h  3xh2  h 3 )  2 x  2h  1
f ( x  h)  2 x 3  6 x 2 h  6 xh2  2h 3  2 x  2h  1
f ( x  h)  f ( x) 2 x 3  6 x 2 h  6 xh2  2h 3  2 x  2h  1  (2 x 3  2 x  1)

h
h
f ( x  h)  f ( x) 6 x 2 h  6 xh2  2h 3  2h h(6 x 2  6 xh  2h 2  2)


h
h
h
f ( x  h)  f ( x )
 6 x 2  6 xh  2h 2  2
h
What is a piecewise function?
• A function that is defined differently for different
parts of the domain.
• Examples: You are paid $10/hr for work up to 40
hrs/wk and then time and a half for overtime.
f ( x)  {10 x, x  40;15 x, x  40}
Increasing and Decreasing
Functions
• Increasing: Graph goes “up” as you move
from left to right.
x1  x2 , f ( x1 )  f ( x2 )
• Decreasing: Graph goes “down” as you move
from left to right.
x1  x2 , f ( x1 )  f ( x2 )
• Constant: Graph remains horizontal as you
move from left to right.
x1  x2 , f ( x1 )  f ( x2 )
Even & Odd Functions
• Even functions are those that are mirrored
through the y-axis. (if –x replaces x, the y
value remains the same) (i.e. 1st quadrant
reflects into the 2nd quadrant)
• Odd functions are those that are mirrored
through the origin. (if –x replaces x, the y
value becomes –y) (i.e. 1st quadrant
reflects into the 3rd quadrant)
Determine if the function is even,
odd, or neither.
f ( x)  2( x  4)  2 x
2
1. Even
2. Odd
3. Neither
2
1.4
• Linear Functions & Slope
Objectives
• Calculate a line’s slope.
• Write point-slope form of a line’s equation.
• Write & graph slope-intercept from of a
line’s equation.
• Graph horizontal or vertical lines.
• Recognize & use the general form of a
line’s equation.
• Use intercepts to graph.
• Model data with linear functions and
predict.
What is slope? The steepness of
the graph, the rate at which the y
values are changing in relation to
the changes in x.
How do we calculate it?
y y2  y1
slope  m 

x x2  x1
A line has one slope
• Between any 2 pts. on the line, the slope MUST
be the same.
• Use this to develop the point-slope form of the
equation of the line.
y  y1  m( x  x1 )
• Now, you can develop the equation of any line if
you know either a) 2 points on the line or b) one
point and the slope.
Find the equation of the line that
goes through (2,5) and (-3,4)
• 1st: Find slope of the line
m= 5  4  1
2  (3)
5
2nd: Use either point to find the equation of the line &
solve for y.
1
y  5  ( x  2)
5
1
2
1
3
y  x 5  x4
5
5
5
5
Slope-Intercept Form of the
Equation of the Line
• It is often useful to express the line in
slope-intercept form, meaning that the
equation quickly reveals the slope of the
line and where it intercepts the y-axis.
• It is REALLY use of the point-slope form,
except that the point is the intercept, (0,b).
• y - b=m(x - 0) becomes y = mx + b
• This creates a quick equation to graph.
Horizontal & Vertical lines
• What is the slope of a vertical line?
– It is INFINITELY steep, it only rises. It is SO steep, we
can’t define it, therefore undefined slope.
– Look at points on a line, i.e. (-4,8),(-4,9),(-4,13),(-4,0).
We don’t care what value y has, all that matters is that
x= - 4. Therefore, that is the equation of the line!
• What is the slope of a horizontal line?
– There is no rise, it only runs, the change in y is zero, so
the slope = 0.
– Look at points on the line, i.e., (2,5), (-2,5), (17,5), etc.
We don’t care what value x has, all that matters is that
y = 5. Therefore, that is the equation of the line!
1.5 More on SLOPE
• Objectives
– Find slopes & equations of parallel &
perpendicular lines
– Interpret slope as a rate of change
– Find a function’s average rate of change
Parallel lines
• Slopes are equal.
• If you are told a line is parallel to a given
line, you automatically know the slope of
your new line (same as the given!).
• Find the equation of the line parallel to
y=2x-7 passing through the point (3,-5).
– slope = 2, passes through (3,-5)
y - (-5) = 2(x – 3)
y + 5 = 2x – 6
y = 2x - 11
Perpendicular lines
• Lines meet to form a
right angle.
• If one line has a very
steep negative slope,
in order to form a right
angle, it must
intersect another line
with a gradual
positive slope.
• The 2 lines graphed
here illustrate that
relationship.
• What about the intersection of a horizontal line
and a vertical line? They ALWAYS intersect at a
right angle. Since horizontal & vertical lines are
neither positive or negative, we simply state that
they are, indeed, ALWAYS perpendicular.
• What about all other lines? In order to be
perpendicular, their slopes must be the negative
reciprocal of each other. (HINT: think about the
very steep negative-sloped line perpendicular to
the gradual positive-sloped line)
1
m1 
m2
Find the equation of the line
perpendicular to y = 2x – 7 through
the point (2,7) .
1.
2.
3.
4.
y=½x+7
y = - 2 x + 1/7
y = - ½ x + 4/3
y=-½x+8
Average Rate of Change
• Slope thus far has referred to the change
of y as related to the change in x for a
LINE.
• Can we have slope of a nonlinear
function?
• We CAN talk about the slope between any
2 points on the curve – this is the
average rate of change between those 2
points!
1.6 Transformation of Functions
•
•
•
•
•
Recognize graphs of common functions
Use vertical shifts to graph functions
Use horizontal shifts to graph functions
Use reflections to graph functions
Use vertical stretching & shrinking to graph
functions
• Use horizontal stretching & shrinking to
graph functions
• Graph functions w/ sequence of
transformations
• Vertical shifts
– Moves the graph up or
down
– Impacts only the “y”
values of the function
– No changes are made
to the “x” values
• Horizontal shifts
– Moves the graph left
or right
– Impacts only the “x”
values of the function
– No changes are made
to the “y” values
Recognizing the shift from the
equation, examples of shifting the
2
function f(x) = x
• Vertical shift of 3 units up
f ( x)  x , h( x)  x  3
2
2
• Horizontal shift of 3 units left (HINT: x’s go the
opposite direction that you might believe.)
f ( x)  x , g ( x)  ( x  3)
2
2
Combining a vertical & horizontal
shift
• Example of function
that is shifted down 4
units and right 6 units
from the original
function.
f ( x)  x , g ( x )  x  6  4
Reflecting
• Across x-axis (y becomes negative, -f(x))
• Across y-axis (x becomes negative, f(-x))
Horizontal stretch & shrink
• We’re MULTIPLYING
by an integer (not 1 or
0).
• x’s do the opposite of
what we think they
should. (If you see 3x
in the equation where
it used to be an x, you
DIVIDE all x’s by 3,
thus it’s compressed
horizontally.)
g ( x)  (3 x)  4
2
VERTICAL STRETCH (SHRINK)
• y’s do what we think
they should: If you
see 3(f(x)), all y’s are
MULTIPLIED by 3 (it’s
now 3 times as high
or low!)
f ( x)  3 x  4
2
Sequence of transformations
• Follow order of operations.
• Select two points (or more) from the original function and
move that point one step at a time.
f ( x)  x 3
3 f ( x  2)  1  3( x  2)  1
3
f(x) contains (-1,-1), (0,0), (1,1)
1st transformation would be (x+2), which moves the function left
2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (1,1)
2nd transformation would be 3 times all the y’s, pts. are now (3,-1), (-2,0), (-1,3)
3rd transformation would be subtract 1 from all y’s, pts. are now
(-3,-2), (-2,-1), (-1,2)
Graph of Example
f ( x)  x 3
g ( x)  3 f ( x  2)  1  3( x  2)3  1
1.7 Combinations of Functions;
Composite Functions
• Objectives
– Find the domain of a function
– Combine functions using algebra.
– Form composite functions.
– Determine domains for composite functions.
– Write functions as compositions.
Using basic algebraic functions,
what limitations are there when
working with real numbers?
• A) You canNOT divide by zero.
– Any values that would result in a zero
denominator are NOT allowed, therefore the
domain of the function (possible x values)
would be limited.
B) You canNOT take the square root (or any
even root) of a negative number.
Any values that would result in negatives
under an even radical (such as square roots)
result in a domain restriction.
Example
• Find the domain
x2
2
x  5x  6
• There are x’s under an even radical AND x’s in the
denominator, so we must consider both of these
as possible limitations to our domain.
x  2  0, x  2
x 2  5x  6  0
( x  3)( x  2)  0, x  2,3
Domain : {x : x  2, x  3}
Algebra of functions
• (f + g)(x) = f(x) + g(x)
• (f - g)(x) = f(x) – g(x)
• (fg)(x) = f(x)g(x)
f
f ( x)
( x) 
, g ( x)  0
g
g ( x)
Composition of functions
• Composition of functions means the output from
the inner function becomes the input of the
outer function.
• f(g(3)) means you evaluate function g at x=3,
then plug that value into function f in place of the
x.
• Notation for composition:
( f  g )( x)  f ( g ( x))
1.8 Inverse Functions
• Objectives
– Verify inverse functions
– Find the inverse of a function.
– Use the horizontal line test to deterimine oneto-one.
– Given a graph, graph the inverse.
– Find the inverse of a function & graph both
functions simultaneously.
What is an inverse function?
• A function that “undoes” the original function.
• A function “wraps an x” and the inverse would
“unwrap the x” resulting in x when the 2
functions are composed on each other.
1
1
f ( f ( x))  f ( f ( x))  x
How do their graphs compare?
• The graph of a function
and its inverse always
mirror each other through
the line y=x.
• Example:y = (1/3)x + 2
and its inverse = 3(x-2)
• Every point on the graph
(x,y) exists on the inverse
as (y,x) (i.e. if (-6,0) is on
the graph, (0,-6) is on its
inverse.
Do all functions have inverses?
• Yes, and no. Yes, they all will have
inverses, BUT we are only interested in
the inverses if they ARE A FUNCTION.
• DO ALL FUNCTIONS HAVE INVERSES
THAT ARE FUNCTIONS? NO.
• Recall, functions must pass the vertical
line test when graphed. If the inverse is to
pass the vertical line test, the original
function must pass the HORIZONTAL line
test (be one-to-one)!
How do you find an inverse?
• “Undo” the function.
• Replace the x with y and solve for y.
1.9 Distance & Midpoint Formulas;
Circles
• Objectives
– Find distance between 2 points.
– Find midpoint of a line segment.
– Write standard form of a circle’s equation.
– Give center & radius of a circle whose
equation is in standard form.
– Convert general form of a circle’s equation to
standard form.
Distance formula is an application
of the Pythagorean theorem.
• Recall for any right triangle, the sum of the squares of the
legs equals the square of the hypotenuse.
c  a b
2
2
2
• For any 2 points, you can imagine a rt. Triangle that would
have those 2 pts. The length of the vertical leg is the
difference in the y values. The length of the horizontal leg
is the difference between the x’s.
• Distance between the points= ( x2  x1 ) 2  ( y2  y1 ) 2
Midpoint is the point in the middle,
therefore the x-value is the average
of the 2 given x’s and the y-value is
the average of the 2 given y’s.
• Midpoint of
 x1  x2 y1  y2 
( x1 , y1 ), ( x2 , y2 )  
,

2 
 2
Standard form or a circle with
radius=r and center at (h,k)
( x  h)  ( y  k )  r
2
2
2
• A circle centered at (-2,5) with a radius=7 has
what equation?
( x  2)  ( y  5)  49
2
2
• A circle whose equation is ( x  2) 2  ( y  6) 2  4
has what as its center and radius?
center = (2,-6) and radius=2
What if the equation is not in
standard form?
You may have to complete the
square.
x  y  8 x  6 y  11  0
2
2
( x 2  8 x)  ( y 2  6 y )  13
( x  4) 2  16  ( y  3) 2  9  11
( x  4) 2  ( y  3) 2  11  16  9  36  6 2