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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
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.
2. Graph f(x) = 1/2x + 2 using the x- and y-intercepts.
3. Graph -2x + y = 0
3.3.3 Vertical and 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)
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
2. How many widgets must she sell to break even?
3. Graph the profit function.
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
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.
As was mentioned we wish to describe lines using two numerical
characteristics . One of those is the
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 (x1, y1) and (x2, y2) 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 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. -
1. Consider the equation y = 2/3x + 2 and determine the slope and y-intercept.
2. Write the equation -3x + 4y = 8 in slope-intercept form and determine the
slope and y-
3.4.4 Graphing Linear Equations Using Slope and y-Intercept
EX 19. Graph the following equations using the slope and y-intercept: