Download Section 1-2: Graphs and Lines

Section 1-2: Graphs and Lines
Cartesian Coordinate System
• x-axis
• y-axis
• quadrants I – IV
• ordered pairs: (x, y)
• origin: (0, 0)
Definition: A linear equation in two
variables is an equation that can be written
in the standard form:
Note: No denominators!
Ex:
Characteristics of lines:
• x-intercept
• y-intercept
• slope
How many points are needed to graph a
line equation?
It’s easy to use two special points:
Using intercepts to graph
• x-intercept: the point where the graph
intersects the x-axis. To find x-intercepts,
let = 0 and solve for . Always express
as an ordered pair (x, 0)
• y-intercept: the point where the graph
intersects the y-axis. To find y-intercepts,
let = 0 and solve for . Always express
as an ordered pair (0, y)
Ex graph 2x – 3y = 6.
Ex: graph 4x + 2y = 0
Note:The x- and y-intercept is the same
point – which is not enough to graph a line.
You need another point! Pick any x-value &
plug it in & solve for y.
How to find slope of a line:
EX: Find the slope of the line passing
through the given two points:
a. (-2, 7) and (-3, -1)
b. (8, 2) and (3, 4)
Note: If m > 0 (slope is positive), the line
rises from left to right;
If m < 0 (slope is negative), the line falls
from left to right;
Special lines: Ax + By = C
1) Vertical line: If A ≠ 0, B = 0 (i.e. No yterm)
Ex: x = 2
x-intc:
y-intc:
Note: No matter what the y-value is, x
always equals 2.
m=
Characteristics: x = C/A
• x-intc:
• y-intc:
• Slope:
2) Horizontal Line: If A = 0, B ≠ 0 (i.e.
No x-term)
Ex: y = -1
Characteristics: y = C/B
• x-intc:
• y-intc:
• Slope:
Three forms of a line:
1.
Point-slope form:
Advantage: works with any point, not just yintercept.
2.
Slope-intercept form:
3.
Standard form: Ax + By = C
(A, B, C are integers)
Ex: a) Write an equation of a line w/slope
-3/7 and y-intc (0, 2)
b) Write an equation of a line passing
through (-2, 1) and (0, 5)
Note: An equation in standard form can
easily be put into slope-intercept form by
simply solving for y.
Ex:
Ex: Write an equation of a line w/slope -1/2
going through (3, -2). Give all three forms.
Ex: Find the standard form and slopeintercept form of the equation of the line that
passes through the points (3, -2) and (7, 4)
Example 3 (vertical/horizontal lines) Write
the equation of the vertical and horizontal
lines through the point (6, -5).
Example 4 (Finding information about a
line with graph given)
Use the graph of each line to find xintercept, y-intercept, and slope. Write the
slope-intercept form of the equation of the
line.
Application Problems
Example 6 (cost equation)
The management of a company that
manufactures skateboards has fixed cost
(cost at 0 output) of $300 per day and total
cost of $4,300 per day at an output of 100
skateboards per day. Assume that cost C is
linearly related to output x.
A) Find an equation of the line relating
output to cost. Write the final answer in
the form C = mx + b;
B) Graph the cost equation from part A)
for 0 ≤ x ≤ 200 .
Practice Problems:
1. Find the slope of the line
represented by the equation
3x – 2y = 6
2. Write the equation of the line that
passes through points with
coordinates (4, 2) and (-2, -6).
Write this equation in point slope
form, slope intercept form and
standard form.
Answers to practice problems:
1.
3/2
2. y – 2 = 4/3(x - 4), y = 4/3x – 10/3, 4x
– 3y = 10