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Graphs and Functions
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).
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 xintercept
is the point where the graph crosses the xaxis. The yintercept is the point
where the graph
crosses the yaxis.
•To find the yintercept, set x = 0 and solve for y .
•To find the xintercept, set y = 0 and solve for x.
EX 12. 1. Graph 3x = 6y + 12 using the x and yintercepts.
2. Graph f(x) = 1/2x + 2 using the x and yintercepts.
3. Graph 2x + y = 0
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)
Vertical Lines
Any equation of the form x = a will always be a vertical line.
EX 14. Graph the equation x = 4
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 pro
fits.
2. How many widgets must she sell to break even?
3. Graph the profit function.
3.4 SlopeIntercept 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 yintercept
and the slope.
3.4.1 Understand Translations
Consider the graph of the function y = 1/2x. 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 .
What are the yintercepts? 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).
EX 16. We examine how to find slope:
1. Look at the graphs from the previous example , find the slope of the lines.
2. Graph the equations y = 2x and and y = 2/3x and find their slopes.
The slope of the line through the distinct points (x_{1}, y_{1}) and (x_{2}, y_{2}) is
provided that We usually use the lowercase letter m to denote the slope.
EX 17. Calculate the slope for the following lines:
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 SlopeIntercept Form
The slopeintercept form of a linear equation is
y = mx + b
where m is the slope of the line and (0, b) is the yintercept of the line.
To write an equation in slopeintercept form, solve the equation for y.
EX 18. 
1. Consider the equation y = 2/3x + 2 and determine the slope and yintercept.
2. Write the equation 3x + 4y = 8 in slopeintercept form and determine the
slope and y
intercept.
3.4.4 Graphing Linear Equations Using Slope and yIntercept
EX 19. Graph the following equations using the slope and yintercept:
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