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Chapter 1 Linear Functions and Mathematical Modeling Section 1.1

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Title: Chapter 1 Linear Functions and Mathematical Modeling Section 1.1


1
Chapter 1 Linear Functions and
Mathematical Modeling Section 1.1

2
Section 1.1Linear Equations and Intercepts
  • Intercepts
  • General Form of a Linear Equation
  • Graphing Lines Using the Intercepts
  • Interpreting Intercepts in Different Contexts

3
The graph of a linear equation is a straight
line. The x-intercept of the graph of a line is
the point where the line crosses the x-axis
The y-intercept of the graph of a line is the
point where the line crosses the y-axis.

4
The general (or standard) form of a linear
equation is denoted by ax by c, where a, b,
and c are real numbers, a and b cannot both be
0. These are examples of linear equations
4x 2y 6.2 y 3.7x 5 y 3 These are
not linear equations x2 y2
9
5
Find the x-and-y-intercepts of the linear
equation 3x ¾ y 18.   To find the
x-intercept, set y 0 and solve for x. 3x ¾
(0) 18 3x 18 x
6 or the ordered pair (6, 0) To find the
y-intercept, set x 0 and solve for y. 3(0) ¾
y 18 ¾ y 18 (4/3)(¾) y
18(4/3) y 24 or the ordered
pair (0, 24)
6
Graphing a Line Method 1 Using a table of
values Create a table of values, plot the
ordered pairs, and connect the points with a
straight line.
7
Graph the equation 2x y 6 by using a table
of values. Pick a set of x-values, substitute
them (one at a time) into the equation to find
the corresponding y-value. x 1
2(1) y 6 2 y 6
2 2 y 6 2
y 4 Plug this value into the table and
repeat this process for x 0 and x 2. Plot the
ordered pairs, and connect the points with a
straight line.

x y
-1
0
2
x y
-1 4
0 6
2 10
8
Graphing a Line Method 2 Using the
intercepts Find the x-intercept, find the
y-intercept, plot the intercepts, and connect the
points with a straight line.
9
Use the intercepts to graph 3x 4y
24. x-intercept 3x 4(0) 24 3x 24 x 8
x-intercept is (8, 0) y-intercept 3(0)
4y 24 4y 24 y 6 y-intercept
is (0, 6) Now plot the intercepts, and connect
the points with a straight line.
10
Maria needs to purchase graphing calculators and
engineering paper for her students. She has 3225
in her budget. The calculators cost 129 each,
and the packages of engineering paper cost 15
each. The equation 129x 15y 3225 represents
Maria's purchases. Find and interpret both
intercepts. x-intercept 129x 15(0)
3225 x 25 Maria can buy 25 calculators
and no engineering paper for 3225. y-intercept
129(0) 15y 3225 y 215 Maria
can buy 215 packages of engineering papers and no
calculators for 3225.  
11
Special Note
  • The x-intercept is also known as the horizontal
    intercept.
  • The y-intercept is also known as the vertical
    intercept.

12
  • Interpreting Horizontal and Vertical Intercepts
  • from a Graph
  • To help you interpret the intercepts, first
    determine
  • what is represented by the horizontal axis and
    the
  • vertical axis.

13
After a tropical storm, Jerrys air boat was
flooded with 40 gallons of water and he had to
remove the water with his onboard sump pump. The
graph below depicts the amount of water G (in
gallons), left in the boat after t minutes of
using the sump pump. Use the graph to find and
interpret both intercepts in the context of the
problem. The horizontal axis, t,
represents Time in minutes The vertical axis, G,
represents Gallons of Water The horizontal
intercept is (25, 0). The sump pump will remove
the water from the boat after 25 minutes. The
vertical intercept is (0, 40). Before the sump
pump was used, there were 40 gallons of water in
the boat.    
14
  • Writing equations from
    application problems
  • Luigi has 84 to spend in apps for his iPad. He
    wants to buy action
  • and sports games apps. The action games he likes
    cost 10.50
  • each and the sports games cost 7 each. Let x
    represent number
  • of action games apps and y represent number of
    sport games apps.
  • a. Write an equation that describes what Luigi
    can purchase with
  • his money.
  • Action games at 10.50 each Sports games at 7
    each 84
  • 10.50x 7y 84
  • b. If Luigi only purchases action games apps, how
    many can
  • he buy?
  • Since no sports games apps are bought, then y
    0.
  • 10.50x 7(0) 84
  • x 8

15
  • (Contd.)
  • c. If Luigi only purchases sports games apps, how
    many can
  • he buy?
  • Our equation was 10.50x 7y 84
  • Since no action games apps are bought, then x
    0.
  • 10.50(0) 7y 84
  • y 12 Luigi can buy 12 sports games
    apps.
  • d. Sketch the graph of the linear equation using
    your answers
  • from parts (b) and (c).
  • Using the x-intercept, (8, 0), and the
    y-intercept, (0, 12), we can
  • graph the linear equation

16
  • Viewing Window Notation for Graphing
    Calculator
  • Xmin, Xmax, Xscl by
    Ymin,Ymax,Yscl
  • Given that a line has intercepts (30, 0) and (0,
    125), which of the
  • following viewing windows will be the most
    appropriate?
  • a. 35, 10, 5 by 30, 120, 20
  • b. 50, 40, 10 by 40, 140, 20
  • We are looking for a viewing window that will
    show both intercepts.
  • Only b will display both (30, 0) and (0, 125).
  • Option a will not show the y-intercept, since
    the Y-max is 120.

17
Using your textbook, practice the problems
assigned by your instructor to review the
concepts from Section 1.1.
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