Lecture Note
Complete class note section 1.1.pdf
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University:
Georgia Southern University -
Course:
MATH 1111 | College Algebra Academic year:
2024
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Views:
387
Pages:
11
Author:
Jalen W.
Graphs and Graphing Utilities
Introduction to Graphing:
The coordinate system
y-axis
quadrant II quadrant I
(-,+) (+,+)
x-axis
quadrant III quadrant IV
(-,-) (+,-)
Where the x-axis and y-axis meet is called the origin.
The ordered pair (x, y) is used to plot points. An ordered pair must be alphabetical.
Example (m,n), (s,t)
The origin has the ordered pair (0,0)
(2,3) Start at the origin, move to the right 2 units, then up 3 units; draw point
(-2,3) Start at the origin, move to the left 2 units, then up 3 units; draw point
(-2,-3) Start at the origin, move to the left 2 units, then down 3 units; draw point
(2,-3) Start at the origin, move to the right 2 units, then down 3 units; draw point
(-5/2, 3/2) Start at the origin, move to the left 5/2 units, then up 3/2 units: draw point
(-2 1/2, 1 1/2)
Graphs of Equations
A relationship between two quantities can be expressed as an equation in two variables,
such as y = 4 - x2
An ordered pair is a solution of an equation in two variables if the values of x and y are
plugged into the equation and the simplified equation is a true statement.
Determining if an ordered pair is a solution to the equation:
1) Plug the value for x and y into the equation.
2) Simplify
3) When simplified, if the equation is true then the ordered air is a solution. If the
equation is false, then the ordered pair is not a solution.
y = 2x + 5 (1, 7)
2p - 3q = -13 (4, 2)
7 = 2(1) + 5
7 = 2 + 5
7 = 7
2(4)
8 - 6 = -13
[2 ≠ -13]
The ordered pair (1, 7) is a solution
The ordered pair (4, 2) is not a solution
Section 1.1 Graphs and Graphing Utilities
Page 3
Linear Equation Forms
Ax + By = C Standard form
y = mx + b Slope-intercept form
The highest exponent in a linear equation is one. The graph of a linear equation is a straight line. To graph a straight line, there must be at least two points.
Ways to graph a linear equation:
Use a T-Chart to Plot Points.
Plug values in for one variable and solve for the other. Plot the points and then connect the points with a straight line. Always solve for y first.
Find the x-intercept and y-intercept
Plot the intercepts and then connect the points with a straight line.
Use y = mx + b.
Plot the y-intercept and use the slope to plot other points and then connect the points with a straight line.
(WILL TALK ABOUT IN ANOTHER SECTION)
Using a T-chart to graph:
3x + y = 2
Although two points is the minimum needed to draw a straight line, three values are preferred. Pick a negative number, zero, and a positive number to plug into one variable.
x | y | 3x + y = 2 | 3x + y = 2 | 3x + y = 2
-- | -- | ----------- | ----------- | -----------
-1 | 5 | 3(-1) + y = 2 | -3 + y = 2 | y = 5
0 | 2 | 3(0) + y = 2 | 0 + y = 2 | y = 2
1 | -1 | 3(1) + y = 2 | 3 + y = 2 | y = -1
Plot the points (-1, 5), (0, 2), (1,-1)
3x + 2y = 4
2y = -3x + 4
y = -3/2x + 2
When you select values for x, pick ±2 or multiples of 2 so the fraction reduces to a whole number.
x | y | y = -3/2x + 2 | y = -3/2x + 2 | y = -3/2x + 2
-- | -- | -------------- | -------------- | --------------
-2 | 5 | y = -3/2(-2) + 2 | y = 3 + 2 | y = 5
0 | 2 | y = -3/2(0) + 2 | y = 0 + 2 | y = 2
2 | -1 | y = -3/2(2) + 2 | y = -3 + 2 | y = -1
Plot the points (-2, 5), (0, 2), (2,-1)
Graph the equations. Let x = -3, -2, -1, 0, 1, 2, 3
[y = -2|x|]
x | y
-- | --
-3 | -6
-2 | -4
-1 | -2
0 | 0
1 | -2
2 | -4
3 | -6
[y = -\frac{1}{2}x + 2]
x | y
-- | --
-3 | 3 1/2
-2 | 3
-1 | 2 1/2
0 | 2
1 | 1 1/2
2 | 1
3 | 1/2
Using the x-intercept and y-intercept to graph:
What is an intercept?
The x-intercept is a point on the x-axis.
The y-intercept is a point on the y-axis.
To find the x-intercept, let y = 0 and solve for x.
Example:
3x + y = 2
3x + (0) = 2
3x = 2
x = 2/3
The x-intercept is (2/3, 0)
The x-intercept is at 2/3 on the x-axis.
To find the y-intercept, let x = 0 and solve for y.
Example:
3(0) + y = 2
0 + y = 2
y = 2
The y-intercept is (0,2)
The y-intercept is at 2 on the y-axis
Plot the points (2/3, 0) and (0,2)
(0.6,)
x-intercept: (1, 0)
y-intercept: (0, 2)
x-intercept: (-1, 0) & (1, 0)
y-intercept: (0, 1)
x-intercept: None
y-intercept: (0, 2)
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