Learning Objectives for Section 1'2 Graphs and Lines

1
Learning Objectives for Section 1.2 Graphs and
Lines

• After this lecture and the assigned homework, you
  should be able to
• calculate the slope of a line.
• identify and work with the Cartesian coordinate
  system.
• graph lines using the slope-intercept method.
• graph lines using the graphing calculator.
• graph lines using the intercepts.
• graph special forms of equations of lines.
• write equations of lines given two points.
• solve applications of linear equations.

2
Linear Equations in Two Variables

• A linear equation in two variables is an equation
  that can be written in the standard form Ax By
  C,
• where A, B, and C are constants (A and B not both
  0), and
• x and y are variables.
• Linear equations graph lines. Anything not
  straight is called a curve.
• A solution of an equation in two variables is an
  ordered pair of real numbers that satisfy the
  equation.
• For example, (4,3) is a solution of 3x - 2y 6.
• The solution set of an equation in two variables
  is the set of all solutions of the equation.
• The graph of an equation is the graph of its
  solution set.

3
Linear Equations
How can you tell a linear equation from other
equations?
A linear equation can be written in the forms
1.
GENERAL FORM
where A, B, and C are constants and x and y are
variables.
(numbers)
SLOPE-INTERCEPT FORM
where m is the slope and b is the y-intercept.
POINT-SLOPE FORM
where m is the slope and (x1,y1) is a point.
2.
A linear equation graphs a straight line.
4
Slope of a Line

• Slope of a line

rise
run
Note The slope of a line is the SAME everywhere
on the line!!! You may use any two points on the
line to find the slope.
5
Calculating Slope
Example Calculate the slope between the given
points. a) (-5, -2) and (-4, 11)
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Calculating Slope
Example Calculate the slope between the given
points. b) (-3, 5) and (2, 5)
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Calculating Slope
Example Calculate the slope between the given
points. c) (4, 7) and (4, -3)
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Determining Slope
Estimate the slope of the line graphed below.
y
x
9
Intercepts of a Line
x-intercept where the graph crosses the
x-axis. The coordinates are (a, 0).
To find the x-intercept, let y 0 and solve for
x.
y-intercept where the graph crosses the
y-axis. The coordinates are (0, b).
To find the y-intercept, let x 0 and solve for
y.
10
Graphing Linear Equations

• To graph linear equations, you may use
• A table of values (aka t-chart, x,y chart)
• The x- and y- intercepts
• The graphing calculator
• The slope-intercept method
• (Personally, I find this method most useful)

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Slope-Intercept Form

• The equation
• y mxb
• is called the slope-intercept form of an
  equation of a line. m represents the slope
• b represents the y-coordinate of the
  y-intercept

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Y-intercept of a Line
As mentioned before
y-intercept where the graph crosses the y-axis.
The coordinates are (0, b).
For instance, if b 6, the line has a
y-intercept at (0, 6).
If y 2x 4, the line has a y-intercept at (0,
-4).
If y -5x 9, the line has a y-intercept at
(0, 9).
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Find the Slope and Intercept from the Equation
of a Line
Example Find the slope and y- intercept of the
line whose equation is 5x - 2y 10.
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Find the Slope and Intercept from the Equation
of a Line
Example Find the slope and y- intercept of the
line whose equation is 5x - 2y 10.
Solution Solve the equation for y in terms of
x. Identify the coefficient of x as the slope and
the y intercept as the constant term.
Therefore the slope is 5/2 and the y intercept
is -5.
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Using Slope-Intercept to Graph a Line
Example Now graph the equation 5x - 2y 10
using the slope-intercept method.
y

• Write equation in slope-intercept form.
• Plot y- intercept.
• Plot other points by counting the slope from the
  y-intercept.
• Draw the line.
• Label the line.

x
16
Using Slope-Intercept to Graph a Line
Example Graph the equation 2x - 3y 18 using
the slope-intercept method.
y
x
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Using Intercepts to Graph a Line
Example Graph 2x - 6y 12 by first finding the
intercepts.

• Compute the x- and y- intercepts.
• Plot the intercepts.
• Compute a 3rd point as a check.
• Draw the line.
• Label the line.

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Using Intercepts to Graph a Line
Example Graph 2x - 6y 12 by finding the
intercepts.
y
x
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Using Intercepts to Graph a Line
Graph 2x - 6y 12.
y
x
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Using a Graphing Calculator
Example Graph 2x - 6y 12 on a graphing
calculator and find the intercepts.

• Get the equation in y mx b form. (Solve for
  y.)
• Type the equation into the calculator.
• Hit ? and enter in the right side of the equation.

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Using a Graphing Calculator (continued)
Example continued Graph 2x - 6y 12 on a
graphing calculator and find the intercepts.

• Hit ?. Adjust the window settings, if necessary.

The standard window settings work well for this
example, but other problems will require
adjusting the window.
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Using a Graphing Calculator (continued)
Example continued Graph 2x - 6y 12 on a
graphing calculator and find the intercepts.
4. Find the intercepts.
To find the x-intercept, Hit ? then ? (CALC
menu). Choose 2 Zero. Then answer the
following questions that appear by using the left
and right arrows
Guess? Place the cursor on your guess of the
x-intercept, then hit
Left bound? pick a point to the left of the
x-intercept, then hit
Right Bound? pick a point to the right of the
x-intercept, then hit
?
?
?
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Using a Graphing Calculator (continued)
Example continued Graph 2x - 6y 12 on a
graphing calculator and find the intercepts.
To find the y-intercept, Hit ?. Press 0 then ?.
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Using a Graphing CalculatorExample
Example Given y -0.6x 15.6 find the
intercepts to one decimal place 1) algebraically
and 2)using the graphing calculator.
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Using a Graphing CalculatorExample (continued)
Example Given y -0.6x 15.6 find the
intercepts to one decimal place 2)using the
graphing calculator.
1. Solve for y, if not already done so.
2. Hit ? then type in the equation.
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Using a Graphing CalculatorExample (continued)
Example Given y -0.6x 15.6 find the
intercepts to one decimal place 2)using the
graphing calculator.
3. Hit ? and decide whether or not to adjust the
viewing window .
Looks like well need to adjust that window!
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Using a Graphing CalculatorExample (continued)
Example Given y -0.6x 15.6 find the
intercepts to one decimal place 2)using the
graphing calculator.
4. Hit ? and choose appropriate values for Xmin,
Xmax, Ymin, Ymax and the scale.
Standard window
These settings do allow us to view the intercepts.
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Using a Graphing CalculatorExample (continued)
Example Given y -0.6x 15.6 find the
intercepts to one decimal place 2)using the
graphing calculator.
5. For the x-intercept
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Special Cases

• The graph of xk is the graph of a vertical line
  k units from the y-axis.
• The graph of yk is the graph of the horizontal
  line k units from the x-axis.

Ex x 5
Ex y -3
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Example
Example Graph 1) x -4 2) y 2
y
x
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Point-Slope Form
The point-slope form of the equation of a line is
where m is the slope and (x1, y1) is a given
point. It is derived from the definition of the
slope of a line
Cross-multiply and substitute the more general x
for x2
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Example
Find the equation of the line through the points
(-5, 7) and (4, 16).
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Example
Find the equation of the line through the points
(-5, 7) and (4, 16). Solution
Now use the point-slope form with m 1 and (x1,
x2) (4,16). (We could just as well have used
(-5,7)).
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Interpreting Slope
Lines that increase from left to right have a
positive slope.
y
x
Lines that decrease from left to right have a
negative slope.
y
x
35
Interpreting Slope
Lines that are horizontal have a slope of 0.
y
x
Lines that are vertical have an undefined slope.
y
x
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Parallel Perpendicular Lines
lines that never intersect.
Parallel lines-
They have the same slopes.
Perpendicular lines-
lines that intersect at right angles. (90 degree
angles.)
They have negative reciprocal slopes.
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Application

• Office equipment was purchased for 20,000
  and will have a scrap value of 2,000 after 10
  years. If its value is depreciated linearly,
  find the linear equation that relates value (V)
  in dollars to time (t) in years

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Application

• Office equipment was purchased for 20,000
  and will have a scrap value of 2,000 after 10
  years. If its value is depreciated linearly,
  find the linear equation that relates value (V)
  in dollars to time (t) in years
• Solution When t 0, V 20,000 and when t
  10, V 2,000. Thus, we have two ordered pairs
  (0, 20,000) and (10, 2000). We find the slope
  of the line using the slope formula. The y
  intercept is already known (when t 0, V
  20,000, so the y intercept is 20,000). The
  slope is (2000-20,000)/(10 0) -1,800.
• Therefore, our equation is V(t) - 1,800t
  20,000.

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Supply and Demand

• In a free competitive market, the price of a
  product is determined by the relationship between
  supply and demand. The price tends to stabilize
  at the point of intersection of the demand and
  supply equations.
• This point of intersection is called the
  equilibrium point.
• The corresponding price is called the equilibrium
  price.
• The common value of supply and demand is called
  the equilibrium quantity.

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Supply and DemandExample
Use the barley market data in the following table
to find (a) A linear supply equation of the form
p mx b (b) A linear demand equation of
the form p mx b (c) The equilibrium
point.
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Supply and DemandExample (continued)
(a) To find a supply equation in the form p mx
b, we must first find two points of the form
(x, p) on the supply line. From the table, (340,
2.22) and (370, 2.72) are two such points. The
slope of the line is
Now use the point-slope form to find the equation
of the line p - p1 m(x - x1)p - 2.22
0.0167(x - 340)p - 2.22 0.0167x - 5.678p
0.0167x - 3.458 Price-supply
equation.
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Supply and DemandExample (continued)
(b) From the table, (270, 2.22) and (250, 2.72)
are two points on the demand equation. The slope
is
p - p1 m(x - x1)p - 2.22 -0.025(x -270)p -
2.22 -0.025x 6.75p -0.025x 8.97
Price-demand equation
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Supply and DemandExample (continued)
(c) If we graph the two equations on a graphing
calculator, set the window as shown, then use the
intersect operation, we obtain
The equilibrium point is approximately (298,
1.52). This means that the common value of supply
and demand is 298 million bushels when the price
is 1.52.