Download File - College Algebra Fundamentals

Lines in the Plane
and Slope
(Part 1)
Section P.4
Linear Equation
• The simplest mathematical model for
relating two variables is the linear equation:
y = mx + b
It is called linear because its graph is a LINE.
What is a Linear Equation?
y
A linear equation is an
equation whose graph
is a LINE.
Linear
Not Linear
Note: A line could be horizontal, vertical, or
diagonal
x
Slope – Intercept Form
y = mx + b
y - coordinate
x - coordinate
Slope
• Symbol is m
•Rise/Run = Δy/Δx =
(y2 – y1)
(x2 – x1)
y – intercept
(where the line
crosses the
y-axis)
What is the y-intercept?
What is the y-intercept?
What is the y-intercept?
Steps to Graph an Equation
Using Slope-Intercept Form
1. Find the slope and the y-intercept
2. Plot the point (0, b).
3. Use the slope to locate the second
point on the line.
4. Draw a line through the two points.
These equations are all in
Slope-Intercept Form:
y  2x  1
y  x  4
3
y x2
2
Notice that these equations are all
solved for y.
Just by looking at an equation in this
form, we can draw the line (no tables).
•The constant is the y-intercept.
•The coefficient is the slope.
y  2x 1
y-intercept = 1.
y  x  4
y-intercept = -4.
3
y x2
2
slope = 2.
slope = -1.
y-intercept = -2.
slope = 3/2.
The formula for Slope-Intercept
Form is: y = mx + b
• ‘b’ is the y-intercept.
• ‘m’ is the slope.
On the next three slides we will graph the three
equations:
3
y  2x  1, y   x  4, y  x  2
2
using their y-intercepts and slopes.
y  2x 1
right 1
right 1 up 2
1) Plot the y-intercept
as a point on the y-axis.
The y-intercept = 1.
up 2
2) Plot more points by
counting the slope up
the numerator (down if
negative) and right the
denominator. The
coefficient, m = 2, so the
slope = 2/1.
y  x  4
1) Plot the y-intercept
as a point on the y-axis.
The y-intercept = -4.
down 1
right 1
down 1
right 1
2) Plot more points by
counting the slope up
the numerator (down if
negative) and right the
denominator. The
coefficient, m = -1, so
the slope = -1/1.
3
y x2
2 1) Plot the y-intercept
right 2
up 3
right 2
up 3
as a point on the y-axis.
The y-intercept = -2.
2) Plot more points by
counting the slope up
the numerator (down if
negative) and right the
denominator. The
coefficient, m = 3/2, so
the slope = 3/2.
Sometimes we must solve the
equation for y before we can graph it.
2x  y  3
2x  y  (2x)  (2x)  3
y  2x  3
The constant, b = 3 is the y-intercept.
The coefficient, m = -2 is the slope.
y  2x  3
1) Plot the y-intercept
as a point on the y-axis.
The y-intercept = 3.
down 2
right 1
down 2
right 1
2) Plot more points by
counting the slope up
the numerator (down if
negative) and right the
denominator. The
coefficient, m = -2, so
the slope = -2/1.
Graphing Horizontal & Vertical Lines
y
When you are asked to
graph a line and there is only
ONE variable in the
equation, the line will either
be vertical or horizontal.
x
Graph x = 3
There are no y values in this
equation, x is always 3, and y
can be any other real number.
x=3
Graphing Horizontal & Vertical Lines
y
When you are asked to
graph a line, and there is
only ONE variable in the
equation, the line will either
be vertical or horizontal.
Graph y = –2
There are no x values in this
equation, y is always – 2, and x
can be any other real number.
x
y = –2
Graphing Horizontal & Vertical
Lines
• Summarized:
• Whenever x = a, the graph is a
vertical line (Slope is undefined)
• Whenever y = b, the graph is a
horizontal line (Slope is zero)
Classwork
• Graphing Lines Worksheet
Homework:
Textbook Pg 51 & 53
Exercises:
1 – 5 (all), 10, 35, 36, 38, 40, 49, 50, 52, 54, 58
***For 49 – 58 (only give the equations)