Download Math 102 Lecture Notes Ch. 3.1 Page 1 of 8 3.1 Graphing Lines

Math 102 Lecture Notes Ch. 3.1 3.1 Graphing Lines Using a Table of Values to Graph Lines
A linear equation in two variables is one that can be written in the form
Ax + By = C
( A and B, not both 0)
We can describe the solution set of all ordered pairs of a linear equation by graphing the equation.
To graph a linear equation, we can make a table of values of a few ordered pairs, plot them, and
then draw a line through them that extends across the coordinate plane.
Example (a) Graph 6x – 3y = 12
Step 1: Solve for y.
Step 2: Complete the table of values.
6x – 3y = 12
x
Step 3:
Step 4: Plot the ordered pairs from the
table of values and draw a line through the
points.
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values.
y
Page 1 of 8 Math 102 Lecture Notes Demonstration Problems 1. (a) Complete a table of values and then use them to graph 4x – 2y = 12 x y Practice Problems 1. (b) Complete a table of values and then use them to graph 6x + 3y = 9 x y Ch. 3.1 Answer: 1. (b) . . Page 2 of 8 Math 102 Lecture Notes Ch. 3.1 Using X-­‐Intercepts and Y-­‐Intercepts to Graph Lines The x-­‐intercept is the point at which the line crosses the x-­‐axis. The y-­‐intercept is the point at which the line crosses the y-­‐axis. The x-­‐coordinate of the y-­‐intercept is 0. The y-­‐coordinate of the x-­‐intercept is 0. Example (b) Graph 6x – 3y = 12
Instead of completing a table of values of many ordered pairs, we can find the x-intercept
and y-intercept of the line, plot the two points and draw a line through them.
Step 1: Complete the table of values
(to find the x-intercept and y-intercept)
Step 3: Plot the two points from the table
of values and draw a line through them.
6x – 3y = 12
x
0
x-intercept
y-intercept
y
0
Step 2:
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values
adjusting the scale where necessary.
Page 3 of 8 Math 102 Lecture Notes Ch. 3.1 Graphing horizontal and vertical lines.
Example (c) Graph y = 3.
Step 1: Write y = 3 as Step 3: Plot the values and draw a line through the points. 0x + y = 3 Step 2: Complete the table of values. x
–2
y
–1
0
1
2
Shortcut: Instead of completing a table of values, draw a line for which y is always 3, that is, crossing (perpendicular to) the y axis at 3. This line is horizontal. Example (d) Graph x = –2.
Step 1: Write x = –2 as Step 3: Plot the values and draw a line through the points. x + 0y = –2 Step 2: Complete the table of values. x
y
–2
–1
0
1
2
Shortcut: Instead of completing a table of values, draw a line for which x is always –2, that is, crossing (perpendicular to) the x axis at –2. This line is vertical. Page 4 of 8 Math 102 Lecture Notes Demonstration Problems 2. (a) Find the x-­‐intercept and y-­‐intercept, then use them to graph 2x – 5y = 10 x
y
x-intercept
0
y-intercept
0 3. (a) Graph x = 6 Ch. 3.1 Practice Problems 2. (b) Find the x-­‐intercept and y-­‐intercept, then use them to graph x + 4y = 4 x
y
x-intercept
0
y-intercept
0
3. (b) Graph y = –1 Answers: 2. (b) 4, 1; . 3. (b) . Page 5 of 8 Math 102 Lecture Notes Demonstration Problems 4. (a) Make a table of values and graph. y = 4x 5. (a) Find the x intercept and y intercept and graph –x + 2y = 6 Ch. 3.1 Practice Problems 4. (b) Make a table of values and graph. y = –2x 5. (b) Find the x intercept and y intercept and graph x – y = 4 Answers: 4. (b) . 5. (b) . Page 6 of 8 Math 102 Lecture Notes Ch. 3.1 Using a Table of Values to Graph Equations that Contain Fractions
In making a table of values for a linear equation, we are free to choose any suitable x
values. If the coefficient of x is a fraction, it is best to choose x values that are multiples
of the denominator of the fraction.
Choose x-values that
are multiples of 3.
Example (e) Graph 2x – 3y = 6
Step 1: Solve for y.
Step 2: Complete the table of values.
2x – 3y = 6
x
–6
y
–3
0
3
6
Step 3:
Step 4: Plot the ordered pairs from the
table of values and draw a line through the
points.
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values.
Page 7 of 8 Math 102 Lecture Notes Demonstration Problems 6. (a) Complete a table of values and then use them to graph 3
y = x + 5 4
x y Practice Problems 6. (b) Complete a table of values and then use them to graph 2
y = − x − 2 5
x y Ch. 3.1 Answer: 6. (b) . . Page 8 of 8