Math 233 - Spring 2009
Chapter 3 - Graphs and Functions
3.1
3.1.1
Graphs
The Cartesian Coordinate System
Definition 1. Cartesian Coordinate System - (or Rectangular coordinate system) consists of
two number lines in a plane drawn perpendicular to each other.
x-axis - the horizontal axis is called the x-axis.
y-axis - the vertical axis is called the y-axis.
Origin - the point of intersection of the two axes is called the origin.
On the coordinate system we will be plotting points. To describe the points in this coordinate
system we use and ordered pair of numbers (x, y). The two numbers, x and y are called the
x-coordinate and y-coordinate respectively.
EX 1. Plot the following points on the same set of axes:
A(2, 3) B(0,1) C(-5, 0) D(-2, -1) E(3,-2)
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3.1.2
Graphs
Definition 2. The graph of an equation is an illustration of the set of points whose coordinates
satisfy the equation.
EX 2. 1. Determine whether the following ordered pairs are solutions of the equation y = −2x + 5.
(a) (1, 3)
(b) (2, 3)
2. Graph y = 2x.
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1
3. Graph y = 21 x + 3.
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REMARK 1. For the graphs above:
• The above graphs are called linear because they are straight lines.
• Any equation whose graph is a straight line is called a linear equation.
• They are also called first degree equations because the highest exponent on any of the
variables is 1.
3.1.3
Nonlinear Graphs
Equations whose graphs are not straight lines are called nonlinear equations. The key to graphing
nonlinear equations is to be sure to plot enough points so we can be sure of what it will look like.
EX 3. 1. Graph y = x2 − 1.
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2. Graph y = x2 .
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2
3. Graph y = |x| + 1.
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3.2
Functions
The concept of a function is one of the most important in all of mathematics. We will discuss several
ways of thinking about and defining functions. But first:
EX 4. Suppose you are driving your car at a constant 40 mph. Can we find a correspondence
between the number of hours driven with the distance travelled?
Definition 3. We have the following terminology:
• The set of all possible times driven is called the domain.
• The set of all possible distances travelled is called the range
EX 5. We have the following schematic:
Definition 4. A function is a correspondence between the first set of elements, the domain, and
a second set of elements, the range, such that each element of the domain corresponds to exactly
one element in the range.
EX 6. Consider the following.
1. (Blackboard)
3
2. (Blackboard)
3. Consider children and biological mothers. The correspondence of children to biological mothers
is a function since for each child there is only one biological mother. However the correspondence between biological mothers to children is not a function, since one biological mother
could have multiple children.
An Alternate Definition
Definition 5. A function is a set of ordered pairs in which no first coordinate is repeated.
EX 7. Determine whether the following are functions:
1. {(1, 6), (2, 3), (4, 3), (5, 7)}
2. {(1, 6), (2, 3), (1, 3), (5, 7)}
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3.2.1
The Vertical Line Test
Most of our functions we will have a domain and range that is either the real numbers or a subset
of the real numbers. For such functions we can graph them on the cartesian coordinate system.
The graph of a function is the graph of its set of ordered pairs.
The vertical line test: If a vertical line can be drawn through any part of the graph and the
line intersects another part of the graph, the graph does not represent a function. If a vertical line
cannot be drawn to intersect the graph at more than one point, the graph represents a function.
Stated more simply: if, on the graph of a function, we can draw a line that intersects at more
than one point, it is not a function. If we can’t, it is a function.
EX 8. Determine whether the following are functions:
1. Consider the following graphs:
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(b)
(a)
2. Use the vertical line test to determine whether the following graphs represent functions. Also
determine the domain and range of each function or relation.
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(a)
(b)
3. Critical thinking: Consider the graph (drawn on blackboard) which represents the speed versus
time of a student driving to school in the morning. Describe what might be occuring for this
function.
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3.2.2
Function Notation
Many of the equations we graphed in sections 3.1 were functions. See examples 2 and 3 from section
3.1. We notice that each of them passes the vertical line test. What are the domain and range of
each of them?
In this class, most of the equations we will encounter will be functions. When an equation is
written in terms of x and y we will frequently wrtie the equation in function notation
f (x) read as “f of x”
Warning: This is NOT multiplication.
EX 9. Let’s consider the equation y = 2x + 1
• Notice that the value of y depends on x.
• If we plug in a value for x we get a value for y, different values of x give different values of x.
• We say that y is a function of x.
• In this case, we can substitute f (x) for y. This tells us what the independent variable is. It
explicitly states that the value depends on x.
• Our function becomes f (x) = 2x + 1
• We will use both notations interchangeably.
REMARK 2. We don’t always us the letter f . Sometimes we use different letters for both our
function and our independent variable. For example g(x), h(x), P (t), etc. . .
EX 10. We also will evaluate functions with this notation.
1. If f (x) = 3x2 − 5x + 1 find
(a) f (4)
(b) f (a)
2. Determine each function value: (a) g(−3) for g(t) =
1
t+4
(b) h(6) for h(x) = 2|x − 10|
3. An application: The Celsius temperature, C, is a function of the Fahrenheit temperature, F.
C(F ) =
5
(F − 32)
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Determine the Celsius temperature that corresponds to 131◦ F
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3.3
Linear Functions
3.3.1
Graph Linear Functions
A linear function is a function of the form
f (x) = ax + b
The graph of a linear function is a straight line. Also, for linear functions, the domain is the set of
all real numbers R. Recall: When graphing y = f (x).
EX 11. Graph f (x) = − 21 x + 1
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3.3.2
Intercepts and Standard Form
The standard form of a linear equation is
ax + by = c
where a, b, and c are real numbers and a and b are not both 0.
In this form it is frequently easier to graph the equation using the intercepts. The x-intercept
is the point where the graph crosses the x-axis. The y-intercept is the point where the graph
crosses the y-axis.
• To find the y-intercept, set x = 0 and solve for y.
• To find the x-intercept, set y = 0 and solve for x.
EX 12.
1. Graph 3x = 6y + 12 using the x- and y-intercepts.
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2. Graph f (x) = 12 x + 2 using the x- and y-intercepts.
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3. Graph −2x + y = 0
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3.3.3
Vertical and Horizontal Lines
Horizontal Lines
Any equation of the form y = b will always be a horizontal line.
EX 13. Graph the equation y = 4 (or written f (x) = 4)
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Vertical Lines
Any equation of the form x = a will always be a vertical line.
EX 14. Graph the equation x = 4
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3.3.4
An Application
EX 15. Suppose a store owner sells widgets for $30 each. If her monthly expenses are $3,000,
answer the following:
1. Construct a function that relates the number of widgets sold to the profits.
2. How many widgets must she sell to break even?
3. Graph the profit function.
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3.4
Slope-Intercept Form of a Linear Equation
Our goal in this section will be to completely describe a line using two numbers which reveal certain
characteristics of the line. The characteristics we will use are the y-intercept and the slope.
3.4.1
Understand Translations
Consider the graph of the function y = 21 x. What happens if we add 2 to the right hand side? How
about if I subract 2? Let’s graph the following functions on the same coordinate system.
y
=
y
=
y
=
1
x+2
2
1
x
2
1
x−2
2
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What are the y-intercepts? Each line is parallel to the other, but the new lines are shifted, or
translated, up or down by two.
3.4.2
Slope
As was mentioned we wish to describe lines using two numerical characteristics. One of those is the
slope.
Definition 6. The slope of a line is the ration of the vertical change (or rise) to the horizontal
change (or run).
slope =
rise
vertical change
=
horizontal change
run
EX 16. We examine how to find slope:
1. Look at the graphs from the previous example, find the slope of the lines.
y
=
y
=
y
=
10
1
x+2
2
1
x
2
1
x−2
2
2. Graph the equations y = 2x and and y = 32 x and find their slopes.
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The slope of the line through the distinct points (x1 , y1 ) and (x2 , y2 ) is
slope =
change in y
y2 − y1
=
change in x
x2 − x1
provided that x1 6= x2 . We usually use the lowercase letter m to denote the slope.
EX 17. Calculate the slope for the following lines:
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(a)
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(b)
-
(c)
REMARK 3. From the example we notice the following:
• Lines with positive slope increase as we go from left to right.
• Lines with negative slope decrease as we go from left to right.
• Any horizontal line has zero slope.
• What would the slope of a vertical line be?
3.4.3
Slope-Intercept Form
The slope-intercept form of a linear equation is
y = mx + b
where m is the slope of the line and (0, b) is the y-intercept of the line.
To write an equation in slope-intercept form, solve the equation for y.
EX 18. -
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1. Consider the equation y = 23 x + 2 and determine the slope and y-intercept.
2. Write the equation −3x + 4y = 8 in slope-intercept form and determine the slope and yintercept.
3.4.4
Graphing Linear Equations Using Slope and y-Intercept
EX 19. Graph the following equations using the slope and y-intercept:
1. −2x + 4y = 8
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2. f (x) = − 23 x + 1
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3.5
Point-Slope Form of a Linear Equation
We investigate one more method of expressing a linear equation. For this, we take the perspective
that we know a point on the line and we know the slope, how can we write an equation for the line?
3.5.1
Point-Slope Form
The point-slope form of a linear equation is
y − y1 = m(x − x + 1)
where m is the slope of the line and (x1 , y1 ) is a point on the line.
EX 20. -
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1. Write the equation of the line with slope 4 and passing through the point (−2, 5).
2. Write, in slope-intercept form, the equation of the line that passes through the points (1, 5)
and (3, 9).
3.5.2
Parallel and Perpendicular Lines
Definition 7. • Two lines are parallel when they have the same slope.
• Two lines are perpendicular when their slopes are negative reciprocals.
REMARK 4. For any number a, its negative reciprocal is
−1
a
or − a1 .
EX 21. Some problems involving parallel and perpindicular lines.
1. Suppose (0, 3) and (3, 0) are two points on line 1 also (7, 4) and (9, 2) are points one line 2.
Determine whether line 1 and line 2 are perpendicular or parallel.
2. Consider the equation 3x + y = 7. Find an equation of a line that has y-intercept of 4 and is
(a) parallel to the given line and
(b) perpendicular to the given line.
3. Consider the equation 3y = 2x + 8
(a) Determine an equation of a line that passes through (6, 1) that is perpendicular to the
graph of the given equation. Write the equation in standard form.
(b) Write the equation from (a) using function notation.
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3.6
The Algebra of Functions
There are several ways in which we can combine functions to get new functions. We examine a few.
Warning: We introduce some new notation, the actual mathematics is pretty straightforward but
you must keep the notation clear.
Operations on Functions
If f (x) represents one function, g(x) represents a second function, and x is in the domain of both
functions, then we have the following operations:
Sum of Functions: (f + g)(x) = f (x) + g(x)
Difference of Functions: (f − g)(x) = f (x) − g(x)
Product of Functions: (f · g)(x) = f (x) · g(x)
Quotient of Functions: (f /g)(x) =
f (x)
g(x) ,
provided g(x) 6= 0
EX 22. Evaluate the following:
1. If f (x) = x2 − x + 3 and g(x) = x + 2
(a) (f + g)(x)
(b) (f − g)(x)
(c) (g − f )(x)
2. If f (x) = x2 − 16 and g(x) = x − 4
(a) (f + g)(3)
(b) (f · g)(5)
3.7
(c) (g/f )(10)
Graphing Linear Inequalities
First a brief review: Recall that when we graph an equation in two variables, we are marking on
the Cartesian Coordinate system all points which satisfy the equation:
EX 23. Graph the equation, 2x + 4y = 8.
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We notice that any pair of numbers, (x, y) that satisfies the equation falls on the line. The
important idea is we are graphically indicating which points satisfy the equation.
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3.7.1
Graph Linear Inequalities in Two Variables
A linear inequality is what results when the equal sign in a linear equation is replaced by an
inequality sign.
EX 24. The following are example of linear inequalities:
1. (a) 2x + 4y < 8
(b) 2x + 4y > 8
2. (a) y ≤ x − 1
(b) y ≥ x − 1
Our goal will be two graph all points which satisfy the inequality. How can we do this?
For this notice that when we graph a linear equation it splits the plane into three regions: the
two sides of the line, and the line itself.
Steps to graphing linear inequalities
1. Replace the inequality with an equal sign.
2. Draw the graph of the equation from step 1.
(a) If the original inequality contains ≤ or ≥, draw a solid line.
(b) If the original inequality contains < or >, draw a dashed line.
3. Select any point not on the line drawn in step 2 and check whether the chosen point solves
the original inequality:
(a) If it does, shade the side of the line containing the point
(b) If it does not, shade the side of the line not containing the point.
EX 25. Graph the following linear inequalities:
1. 2x + 4y < 8
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2. y ≥ x − 1
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3. y ≤ − 14 x
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