Download 2-2-guided-notes

Algebra 2
2-2: Linear Equations
Objective 1: I can graph a linear equation in slope-intercept or standard form.
A function whose graph is a line is a ____________________. You can represent a
linear function with a ____________________, such as __________. A solution of a
linear equation is any __________________ (x, y) that makes the equation true. An
equation in the form y  mx  b is said to be in ____________________________.
Recall from algebra 1 that there are different names for the variables x and y:
x
y_______________________
Let’s review the steps to graph a linear equation that you learned in algebra and
geometry.
* Plot _____ on the y-axis. If b is negative, plot it on the _______________ axis,
if b is positive plot it on the ________________ axis.



rise
.
run
If the slope is positive, move _________________________________________
and if the
If the slope is negative, move _________________________________________
Use the slope to get more points on the line. Recall that the slope, m 
Example 1: Graph each equation.
2
a) y  x  3
3
1
b) y   x
5
QC1: Graph each equation.
a) y  x  3
b) y  2 x  3
Sometimes the equations are written as Ax  By  C , which is called _______________
___________. There are two ways to graph an equation that is written in standard form.
1) Put it into slope-intercept form, y  mx  b .
2) Find the x and y-intercepts of the graph. We do this by letting x and y equal _______.
The ____________________ is where the graph crosses the y-axis, at (0, b).
The ____________________ is where the graph crosses the x-axis, at (a, 0)
Example 2: Graph by finding the intercepts: x  y  2
QC2: Graph by finding the intercepts: 3x  2 y  6
QC 2.5: Graph by finding the intercepts: 4 x  2 y  8
In the next objective, we are going to recall how to write equations of lines. To do so, we
will need to remember how to use the slope formula.
Slope =
rise
=
run
Example 3: Find the slope of the line through the points (3, 2) and (-9, 6).
QC 3: Find the slope of the line through each pair of points.
a) (-2, -2) and (4, 2)
b) (0, -3) and (7, -9)
It is important that you are able to quickly identify the slopes of any line.
Chant: Vertical lines are __________________, horizontal ____________.
Practice Problems: Find the slope of the line that contains each pair of points.
1. (3, -1) and (12, -2)
2. (-3, -5) and (7, 0)
3. (-4, 8) and (2, 2)
4. (6, -3) and (6, 1)
5. (3, -1) and (5, -1)
Find the slope of the line in each graph.
7.
8.
9.
10.
6. (9, -3) and (7, 5)
Objective 2: I can write equations of lines.
To write an equation for a line, you need to be given 2 pieces of information.
Case 1)
or
Case 2)
Case 1: Writing an equation given the slope and a point, you need to use the
_________________________:
Example 4: Write an equation in standard form of the line with a slope of 
1
that
2
contains the point (8, -1).
QC 4: Write in standard form the equation of each line.
A) Slope of 2, through (4, -2)
2
B) Slope of  , through (5, 6)
3
Case 2: When you are given 2 points on a line, find the ________ first, then use the
______________________ from Case 1.
Example 5: Write an equation in slope-intercept form for a line that passes through the
points (1, 5) and (4, -1).
QC5: Write in slope-intercept form the equation for each line that passes through each
pair of points.
A) (5, 0) and (-3, 2)
B) (-4, -3) and (-5, 7)
It is very important that you are able to rewrite an equation from standard form into
slope-intercept form. We need this especially to use the graphing calculators.
Ex 6: Rewrite 4 x  3 y  7 in slope-intercept form. Identify the slope and y-intercept.
QC6: Rewrite 3x  2 y  1 in slope-intercept form. Identify the slope and y-intercept.
Objective 3: I can determine whether lines are parallel or perpendicular.
In the graph below, the two lines are ______________. Parallel lines are lines in the
same plane that never _______________.
The equation of the top line is:
The equation of the bottom line is:
What do these two equations have in common?
Slopes of parallel lines: Non-vertical lines are parallel if they have the
____________________ and different ________________.
Example: The equations
and
have the same slope,
____, and different y-intercepts. The graphs of the two equations are parallel.
You can use _____________________ form of the equation of a line to determine
whether the lines are parallel.
1
Example 7: Are the lines y   x  5 and 2 x  6 y  12 parallel? Explain.
3
QC 7: Are the lines  6 x  8 y  24 and y 
3
x  7 parallel? Explain.
4
The lines below are _______________________. Perpendicular lines are lines that
intersect to form a ________________.
The equation of the positive line is:
The equation of the negative line is:
What do you notice about the slopes of these lines?
Slopes of Perpendicular Lines: Two lines are perpendicular if the product of their
slopes is _____. Their slopes will also be ___________________________.
A vertical and horizontal line are also perpendicular.
Example: The equations
and
have opposite reciprocal
slopes,
(the product of the slopes is also -1), so these lines are perpendicular.
Example 8: Are the lines for each pair of equations parallel, perpendicular, or neither?
Explain.
1
y  x4
A)
4
y  4x  2
2
x6
B)
3
 2x  3 y  6
y
C)
3x  5 y  3
 5x  3 y  8
D)
2x  y  5
y  2x  1
Example 9: Write an equation of a line that is parallel to y  3x  5 and goes through the
point (-2, 1).
Example 10: Write an equation of a line that is perpendicular to y  3x  5 and passes
through the point (6, 2).