Download Chapter 3 Class Notes Intermediate Algebra, MAT1033C SI Leader Joe Brownlee

Chapter 3 Class Notes
Intermediate Algebra, MAT1033C
SI Leader Joe Brownlee
Palm Beach State College
Class Notes
3.1
Professor Burkett
SI Leader Joe Brownlee
3.1 – The Rectangular Coordinate System
An ordered pair, a “point”:
Plotted points on a graph:
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Class Notes
3.1
Professor Burkett
SI Leader Joe Brownlee
A linear equation in standard form is written 𝐴𝑥 + 𝐵𝑦 = 𝐶, where A, B, and C are real numbers, and A and B
are not both 0.
Example 1:
Find ordered pairs that satisfy the given equation.
3𝑥 − 4𝑦 = 12
x
0
4
y
-3
0
Step 1: Begin by substituting 0 for 𝑥 in the given equation, then solve for 𝑦.
3(0)-4y=12
-4y=12
y=-3
Step 2: Now substitute 0 for 𝑦 in the given equation, then solve for 𝑥.
3x-4(0)=12
3x=12
x=4
We now have two points, (0, -3) and (4, 0), that we can plot on a graph to draw our line.
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Class Notes
3.1
Professor Burkett
SI Leader Joe Brownlee
Example 2:
Find ordered pairs that satisfy the given equation.
2𝑥 − 5𝑦 = 10
x
0
5
y
-2
0
Step 1: Begin by substituting 0 for 𝑥 in the given equation, then solve for 𝑦.
2(0)-5y=10
-5y=10
y=-2
Step 2: Now substitute 0 for 𝑦 in the given equation, then solve for 𝑥.
2x-5(0)=10
2x=10
x=5
We now have two points, (0, -2) and (5, 0), that we can plot on a graph to draw our line.
Two useful points for graphing are the x- and y-intercepts. The x-intercept is the point, if any, where the line
intersects the x-axis. The y-intercept is the point, if any, where the line intersects the y-axis.
The above two examples show how to find the x- and y-intercepts of a line. When graphing the equation of a
line, find the intercepts as follows:
Let 𝑦 = 0 to find the x-intercept; let 𝑥 = 0 to find the y-intercept.
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Class Notes
3.1
Professor Burkett
SI Leader Joe Brownlee
When graphing horizontal and vertical lines, the equation of the line will indicate which axis will be
intersected:
Example 3:
Graph 𝑦 = 3.
Example 4:
Graph 𝑥 = 1.
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Class Notes
3.1
Professor Burkett
SI Leader Joe Brownlee
When finding the midpoint between two endpoints, use the following:
Example 5:
Find the coordinates of the midpoint of the line segment 𝑃𝑄 with endpoints 𝑃(4, −3) and 𝑄(6, −1).
Step 1: Substitute the given coordinates into the midpoint formula, then simplify if possible.
(
4+6 −3+(−1)
2
,
2
10 −4
(5, −2)
)=
2
)=( 2 ,
Step 2: Therefore, the midpoint of line segment 𝑃𝑄 is (5, −2).
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Class Notes
3.2
Professor Burkett
SI Leader Joe Brownlee
3.2 – The Slope of a Line
The slope of a line simply describes a line’s steepness, also known as pitch or grade.
This sign, used in real life, indicates a
steep slope so truck drivers know to
slow down when going downhill.
To find the slope 𝒎 of the line through the distinct points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) use the formula:
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Class Notes
3.2
Professor Burkett
SI Leader Joe Brownlee
Example 1:
To find the slope of the line shown on the graph on the previous page with points (1,1) and (5,-2), simply plug
the points into the slope formula and solve:
Step 1: Plug points into slope formula.
−2−1
−3
−3
( 5−1 )= ( 4 )= 4
Step 2: So the slope of the line shown on the graph is
−3
4
.
The slope of a horizontal line is 0.
The slope of a vertical line is undefined.
To find the slope of a line when given an equation, we must put the equation into slope-intercept form, which
is 𝑦 = 𝑚𝑥 + 𝑏, where 𝒎 is the slope and 𝒃 is the y-intercept (0,b).
Example 2:
Given the equation 3𝑥 − 4𝑦 = 12, find the slope of the line and graph.
Step 1: Take the given equation, which is written is standard form, and put it into the form y=mx+b
3x-4y=12
-4y=-3x+12
3
𝑦 = 4𝑥 − 3
3
Step 2: Therefore, 4 is the slope of the line.
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Class Notes
3.2
Professor Burkett
SI Leader Joe Brownlee
Step 3: Now that our line is in y=mx+b form, we can quickly and easily graph our line. Since b in this equation
is -3, we know this line crosses the y-intercept at (0,-3). Go to that point on a graph and draw a dot. From
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that dot, since our slope is 4, we will go up 3 units and to the right 4 units.
The slopes of parallel lines are the exact same.
The slopes of perpendicular lines are opposite reciprocals.
3.2
review
at a
glance:
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Class Notes
3.3
Professor Burkett
SI Leader Joe Brownlee
3.3 – Linear Equations in Two Variables
When given the slope and y-intercept of a line and asked to write the corresponding equation, use the form
y=mx+b.
Example 1:
Write an equation of the line with slope -4 and y-intercept (0,8).
𝑦 = −4𝑥 + 8
Point-Slope Form
The point-slope form of the equation of a line with slope 𝒎 passing through the point (𝒙𝟏 , 𝒚𝟏 ) is:
; m=25
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Class Notes
3.3
Professor Burkett
SI Leader Joe Brownlee
Example 2:
Therefore, the equation for the line passing through the above given points (3, -1) with a slope of 2 is: 𝑦 =
2𝑥 − 7.
Parallel and Perpendicular Lines
When given two equations of lines and asked whether they are parallel or perpendicular, you must first put
both lines in y=mx+b and then compare their slopes.
Notice that since the slopes, m, are the same, they
are parallel.
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Class Notes
3.3
Professor Burkett
SI Leader Joe Brownlee
Notice that since the slopes are opposite (one is
negative, the other positive) reciprocals (the
numerator and denominator have flipped), the
lines are now perpendicular.
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Class Notes
3.4
Professor Burkett
SI Leader Joe Brownlee
3.4 – Linear Equations in Two Variables
The solution of a linear inequality in two variables like Ax + By > C is an ordered pair (x, y) that produces a true
statement when the values of x and y are substituted into the inequality.
Example 1:
Is (1, 2) a solution to the given inequality?
The graph of an inequality in two variables is the set of points that represents all solutions to the inequality.
A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents
the solutions of the inequality.
Graphing a Linear Inequality
Step 1: Draw the graph of the straight line that is the boundary. The boundary line is dashed for > and < and
solid for ≤ and ≥. The half-plane that is a solution to the inequality is usually shaded.
Step 2: Choose a test point. Choose any point not on the line, and substitute the coordinates of this point in
the inequality.
Step 3: Shade the appropriate region. Shade the region that includes the test point if it satisfies the original
inequality. Otherwise, shade the region on the other side of the boundary line.
Example 2:
Graph the inequality
Since the graph is shaded
above the line, we know
any point we choose to substitute
into the original inequality from
that shaded region will make the
inequality true.
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Class Notes
3.4
Professor Burkett
SI Leader Joe Brownlee
Graphing the Intersection of Two Inequalities
Sometimes you will be asked to graph two inequalities on the same grid.

If the inequalities you are graphing are separated by “AND” your answer will be only the section of the
graph that is doubled shaded.

If the inequalities you are graphing are separated by “OR” your answer will be any part of the graph
that is shaded.
Example of an AND graph (just the portion of the graph that is doubled shaded is your answer):
Example of an OR graph (all shaded areas are your answer):
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Class Notes
3.5
3.5 – Intro to Relations and Functions
Relation: an ordered pair: (x,y)
Function: a relation in which for every x, there is only one y.
Example 1:
x
0
-1
-1
y
5
4
3
This is NOT a function, because x has two possible y-values, 3 or 4.
Example 2:
x
0
-1
-3
y
1
3
5
This IS a function, because for every x, there is only one y.
Domain: the set of all x-values
Range: the set of all y-values
Example 3:
(4, 0), (4, 1), (4, 2)
The domain is {4}
The range is {0, 1, 2}
You can determine if a graph is a function by
using a simple test called the vertical line
test. If every vertical line you draw on a
graph intersects the relation (line) in no more
than one point, then the relation is a
function. None of the graphs to the right are
functions, because they fail our vertical line
test.
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Professor Burkett
SI Leader Joe Brownlee
Class Notes
3.5
You can determine the domain and range of a graph, too:
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Professor Burkett
SI Leader Joe Brownlee
Class Notes
3.6
3.6 – Function Notation and Linear Functions
In function notation, 𝑦 = 𝑓(𝑥)
𝑓(𝑥) is pronounced “f of x” or “f at x.”
Example 1:
𝑦 = 9𝑥 − 5 is the same as 𝑓 (𝑥 ) = 9𝑥 − 5
Example 2:
Let 𝑓(𝑥 ) = 6𝑥 − 2. Evaluate the function 𝑓 for each of the following:
(a) 𝑥 = −2
𝑓 (𝑥 ) = 6𝑥 − 2
𝑓 (−2) = 6(−2) − 2
𝑓 (−2) = −12 − 2
𝑓 (−2) = −14
Therefore, (-2, -14) is the ordered pair that belongs to 𝑓.
Example 3:
Let 𝑔(𝑥 ) = 5𝑥 − 1. Find and simplify 𝑔(𝑚 + 2).
𝑔(𝑥 ) = 5𝑥 − 1
𝑔(𝑚 + 2) = 5(𝑚 + 2) − 1
𝑔(𝑚 + 2) = 5𝑚 + 10 − 1
𝑔(𝑚 + 2) = 5𝑚 + 9
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Professor Burkett
SI Leader Joe Brownlee
Class Notes
3.6
Professor Burkett
SI Leader Joe Brownlee
Example 4:
For each function, find 𝑓(−2).
(a) 𝑓 = { (0, 5), (−1, 3), (−2, 1) }
Look for the ordered pair that has a -2 for the x-coordinate. The corresponding y-coordinate will be
your answer.
Therefore, 𝑓(−2) = 1
When you’re asked to graph a function, all you do is treat the function notation ( "𝑓(𝑥)" or "𝑔(𝑥)" ) as a “y”
and plot the line as usual.
Example 5:
Graph the function and give the domain and range.
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𝑓 (𝑥 ) = 4 𝑥 − 2
𝑦=
3
𝑥−2
4
D= (−∞, ∞)
R= (−∞, ∞)
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