Algebra 1 Chapter 4: Graphing Linear Equations and Functions

1
Algebra 1Chapter 4 Graphing Linear Equations
and Functions
Ymxb
X-axis
Y-axis
2
The Coordinate Plane
The origin occurs where the x-axis and the
y-axis intersect at (0, 0)
The x-axis is the horizontal axis on a
coordinate plane
4
3
2
1
1 2 3 4 5
-5
-4 -3 -2 -1
-1
-2
-3
-4
The y-axis is the vertical axis on a coordinate
plane
3
Quadrants of the Coordinate Plane
Coordinate planes are divided up into four
quadrants
4
Quadrant I (, )
Quadrant II (--, )
3
2
1
1 2 3 4 5
X-axis
-5
-4 -3 -2 -1
-1
-2
Quadrant IV (, --)
Quadrant III (--, --)
-3
-4
Y-axis
4
Ordered Pairs
An ordered pair consists of an x-coordinate and a
y-coordinate, usually surrounded by parenthesis.
(x, y)
The x-coordinate is always the first number.
The y-coordinate is always the second number.
Example Plot the point (3, -2)
Starting at the origin, move 3 units right and
then 2 units down.
x
.
(3, -2)
y
5
Standard Form
A linear equation is written in the form AxByC,
where neither A nor B are zero.
A solution of an equation is an ordered pair (x,
y) that makes the equation true.
Example Determine whether (4, -2) is a solution
of 2x3y2
Step 1 plug in 4 for x and -2 for y. (Remember
that x is the first number in an ordered pair
and y is the second.)
2(4)3(-2)2
8(-6)2
Step 2 multiply and simplify.
22
(Be sure to pay attention to any negative signs.)
Since both sides are equal, (4, -2) is a solution
of 2x3y2
6
Finding Solutions of Linear Equations
The first step when finding solutions is to
REWRITE the equation.
Write original equation
4x-2y10
-2y-4x10
Move x to the other side of the equation
y2x-5
Divide by -2 to get your rewritten equation
The next step is to CHOOSE x-values for your
solutions and make a table.
x
-3 -2 -1 0 1 2 3
y
-11 -9 -7 -5 -3 -1 1
Then PLUG IN the values for x in the new equation
to get the y-values.
All of the ordered pairs on the table are
solutions for the given equation.
7
Graphing Linear Equations Using Tables
By finding solutions for an equation, you are
essentially finding points on the graph of that
equation.
Example Graph y-2x-1 by using a table of values.
y2x-1
Rewrite the equation
x
-3 -2 -1 0 1 2
3
Choose x-values, make a table, and plug the
x-values into the equation to find the y-values
-7 -5 -3 -1 1 3 5
y
.
.
Plot the points on the Coordinate plane and
connect the dots
.
.
.
.
.
8
Graphing Horizontal and Vertical Lines
Graphing an equation such as x2 or y-1 is
actually easier than it seems.
Similarly, when an equation is y-1, it means
that for every x-value, the y-value is -1,
making it a HORIZONTAL line.
When an equation is x2, it simply means that
for every y-value, the x-value is 2, making it a
VERTICAL line.
.
.
.
.
.
.
.
x
.
x
.
.
y
y
9
X- and Y-Intercepts
An x-intercept is the x-coordinate of a point
where a graph crosses the x-axis.
.
.
A y-intercept is the y-coordinate of a point
where a graph crosses the y-axis.
A line that is neither horizontal nor vertical
has exactly one x-intercept and one y-intercept.
10
Finding Intercepts
An x-intercept occurs where y0, so to find the
x-intercept, plug in 0 for y in the given
equation.
Example Find the x-intercept of 7y-3x21
7(0)-3x21
-3x21
x-7
The x-intercept for this equation is -7.
A y-intercept occurs where x0, so you can plug
in 0 for x in the equation to find the
y-intercept.
Example Find the y-intercept of 7y-3x21
7y-3(0)21
7y21
y3
The y-intercept for this equation is 3.
11
Graphing Using Intercepts
Once you find the intercepts of an equation, you
can graph that equation by plotting the
intercepts and connecting the two points.
Example Plot the graph of the equation that has
the intercept points (3,0) and (0,-2).
.
.
12
Slope of a Line
To find the slope of a line, its necessary to
divide the rise by the run.
Vertical rise2
Horizontal run4
Vertical rise
2
1
Slope


Horizontal run
4
2
13
Finding Slope
.
(x , y )
.
2 2
y -y
(x , y )
2 1
1 1
x -x
2 1
The slope m of a line that passes through the
points above is
y -y
rise
change in y
2 1
m


run
Change in x
x -x
2 1
14
Positive and Negative Slope
If a line moves down and to the right, then it
has a negative slope.
Example Let two points on a line be (1, 4) and
(7, 3). Find the slope.
3-4
-1

m
7-1
6
If a line moves up and to the right, then it has
a positive slope.
Example Let two points on a line be (6, -2) and
(9, 7). Find the slope.
7-(-2)
9

3
m
9-6
3
15
Zero and Undefined Slope
The slope of any horizontal line is always zero.
Example Let two points on a line be (5, -2) and
(3, -2). Find the slope.
-2-(-2)
0

m
Zero divided by any number is 0. Slope is 0.
3-5
-2
The slope of any vertical line is always
undefined.
Example Let two points on a line be (3, -1) and
(3, 4). Find the slope.
4-(-1)
5
m

Cannot divide by zero, so slope is undefined.
3-3
0
16
Direct Variation
When two quantities y and x have a constant ratio
k, they are said to have direct variation.
If y/x k, then ykx.
Example If variables x and y vary directly and
one pair of values is y24 and x3, write an
equation that relates x and y.
ykx
Write the model for direct variation.
24k(3)
Plug in the x and y values.
8k
Divide to find the value of k.
By plugging the value of k back into the original
model, an equation that relates x and y is y8x.
17
Slope-Intercept Form
The linear equation ymxb is written in
slope-intercept form, where m is the slope and b
is the y-intercept.
Example Graph the equation y2x-4.
Step 1 Recognize that the slope is 2 and the
y-intercept is -4.
Step 2 Plot the point (0,b) where b is -4.
Step 3 Use the slope to locate a second point on
the line.
2
rise
m

Move 2 units down, 1 unit right.
1
run
18
Parallel Lines
Parallel lines are different lines in the same
plane that never intersect.
Two nonvertical lines are parallel if they have
the same slope and different y-intercepts. Any
two vertical lines are parallel.
The red lines are parallel to each other because
they have the same slope.
The blue lines are parallel to each other
because they are vertical.
19
Functions and Relations
A function is a rule that establishes a
relationship between two quantities, called the
input and the output, where for each input,
there is exactly one output.
A relation is any set of ordered pairs. A
relation is a function if for every input there
is exactly one output.
Example Determine whether the relation is a
function.
Input 1 2 3 4
Output 3 5 7
The relation is not a function because the input
1 has two outputs.
20
Vertical Line Test for Functions
A graph is a function if no vertical line
intersects the graph at more than one point.
Function
Not a function
21
Homework

• Slope-intercept form lecture tomorrow!
• Complete Study Guide for Chapter 4
• Due Tuesday, November 19th
• Chapter 4 Test ?
• Wednesday, November 20th