Transcript 11 graphing

GRAPHING LINEAR
EQUATIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
GRAPHING LINEAR
EQUATIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
Basic Definitions
Axes – perpendicular number lines
• x-axis – horizontal number line
• y-axis – vertical number line
• Origin – the point of intersection of the
axes
Ordered pair – a number pair (x, y),
coordinate, point
• Abscissa – first coordinate, x
• Ordinate – second coordinate, y
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
Graphs
On a number line, each point is the graph of a number
0
2
On a plane, each point is the graph of a number pair
ordered pair: (x, y)
(1,2)
Ex: Plot (3, 5)
Ex: Plot (-4, -2)
Ex: Plot (-3, 4)
An ordered pair is a solution to a 2-variable equation
if a true statement results when the equation is
evaluated at the ordered pair.
Ex: Show (4, 5) and (-2, 2) are solutions to y = ½ x + 3
y=½x+3
y=½x+3
5
5
½ (4) + 3
2+3
5
5
2
2
2
True  (4, 5) is a solution!
½ (-2) + 3
-1 + 3
2
True  (-2, 2) is a solution!
Ex: Plot (4, 5) and (-2, 2) on the same set of axes.
Notice a straight line connect the points (solutions)
 Connecting all the points that are solutions to
an equation will result in a straight line.
 In other words, the graph of the line connecting
the points (solutions) represents the solution set of
the equation.
 Since the graph of the solution set is
represented by a straight line, the equation is
called a linear equation.
 More formally, an equation in 2 variables, where
the exponents of the variables are 1, is called a
linear equation.
Ax + By = C or y = mx + b, where A, B, C, m,
and b are constants and A and B are not both 0
Graphing a Linear Equation
(Plotting Points)
1. Solve the equation for one of the
variables, usually y
2. Pick a value for x, plug it in, & solve for y
3. Repeat at least two more times
4. Plot the points on the same set of axes
5. Connect the dots with a straight line
Ex: Graph 6x – 3y = 3
Solve for a variable: y
6x – 3y = 3
-6x
-6x
- 3y = 3 – 6x
-3
-3
y = - 1 + 2x
y = 2x - 1
x
y = 2x - 1
y
0
y = 2(0) - 1
-1
1
y = 2(1) - 1
1
½
y = 2(½) - 1 0
We have identified 3 solutions to the equation:
(0, -1)
(1, 1)
( ½ , 0)
Plotting the three solutions/points we get:
(0, -1)
(1, 1)
( ½ , 0)
The solution points lie on a straight line.
Every point on this line is a solution to the equation
6x – 3y = 3!
1
1
Your turn to try a few
• We can always plot points to graph linear
equations
• However, plotting points could be tedious,
(especially for “messy” equations)
• There must be other ways to plot linear
equations . . .
Graphing using Intercepts
• Consider a linear equation of the form
Ax + By = C
• The y-intercept is the point in which the
graph of the line crosses the y-axis, (0, b)
To find the y-intercept, let x = 0 and solve
for y
• The x-intercept is the point in which the
graph of the line crosses the x-axis, (a, 0)
To find the x-intercept, let y = 0 and solve
for x
Ex: Graph 2x – 3y = 6 using intercepts
y-int: let x = 0
x-int: let y = 0
2(0) – 3y = 6
2x – 3(0) = 6
– 3y = 6
2x = 6
y=-2
x=3
(0, - 2)
1
1
(3, 0)
Your turn to try a few
Slope-Intercept Form
Consider the linear equation Ax + By = C
Solving for y, we get an equation of the form
y = mx + b, where m and b are constants
y = mx + b is called the slope-intercept form because
b is the “intercept” (y-intercept (0, b)) and
m is the “slope”
When the slope is negative (m < 0), the line
slants down from left to right
When the slope is positive (m > 0), the line
slants up from left to right
Graphing using the slope and
y-intercept
Given the slope-intercept form, we can identify
the slope, m, and the y-intercept, (0, b)
To graph an equation of a line, given the slopeintercept form, start by plotting the y-intercept
Then use the slope to identify at least 2 more
solutions of the equation (i.e. solution points)
Recall, y = mx + b, where m and b are numbers,
is the slope-intercept form of a linear equation.
Ex: Find the slope-intercept form of x + 5y = 10
To find the slope-intercept form,
we need to solve for y
y=
mx+b
x + 5y = 10
m = - 1/5
-x
-x
b=2
5y = -1x + 10
(0, 2) is the y-int
5
5
y = (-1/5) x + 2
Graph using Slope-Intercept form: y = (-1/5) x + 2
-1
Slope m =
5
Plot (0, 2)
and y-int = (0, 2)
Next, use the slope rise = - 1  down 1
run = 5  right 5
2
Note: -a/b = a/(-b) 
m = 1/(-5)  up 1, left 5
2
Ex: Graph 2x + 3y = -9
solve for y to use the slope-int form
2x + 3y = - 9
-2x
-2x
3y = - 2x - 9
3
3
y = (-2/3)x – 3
y=
mx+b
m=
3
Negative slope  line slants
down from left to right
rise = -2  down 2
b=-3
-2
run = 3  right 3
 (0, -3) is the y-int
 START HERE
Graph: y = (-2/3)x - 3
-2
Slope m =
3
and y-int = (0, -3)
Next, slope rise = -2  down 2
or rise = 2  up 2
Plot (0, -3)
run = 3  right 3
run = -3  left 3
2
2
Your turn to try a few
Special Lines
• The graph of y = b is a horizontal line with
y-intercept (0, b)
y is always b no matter what x is
• The graph of x = a is a vertical line with
x-intercept (a, 0)
x is always a no matter what y is
Ex: Graph 7x + 63 = 0
7x + 63 = 0
7x = - 63
x=-9
x is always – 9
no matter what y
is  vertical
3
3
Ex: Graph 12y = 48
12y = 48
y=4
y is always 4 no matter what
x is  horizontal
3
3
Horizontal lines have slope m = 0
No vertical change  rise = 0  m = 0
Vertical lines have undefined slope,
m = undefined
No horizontal change  run = 0  m undefined
m=
rise
run
To Graph a Linear Equation:
• Plot points: pick nice x and solve for y  (x, y),
find at least 3 solutions
• Intercepts:
o y-int: set x = 0, solve for y  (0, y)
o x-int: set y = 0, solve for x  (x, 0)
• Slope-Intercept: y = mx + b
o Start with y-int (0, b)
o Use slope m to plot at least 2 more points
• Only one variable:
o y = b  horizontal line with y-int (0, b)
o x = a  vertical line with x-int (a, 0)