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Chapter 5.1 Identify Linear Functions and their Graphs
Given a set of ordered pairs, a constant rate of change means that if the value of x changes by 5 units
the value of y will change also have a constant change. Example: (5, 300), (10, 600), (15, 900), (20,
1200). A constant rate of change can be indicated by the ratio of change in y/ change in x. In our
example the constant rate of change is 300/5. A constant rate of change indicates a linear function. A
linear function is a function whose graph is a non-vertical line. Why can't a linear function be a vertical
line?
To find the change in x you subtract each x value from its succeeding x value. To find the change in y
you would do the same thing.
To determine if a given set of ordered pairs or a relation represents a linear function you need to find
whether there is a constant rate of change. If the change in x and the change in y stays the same
between each set of ordered pairs then you have a linear function.
You can also graph the ordered pairs on a coordinate plane to determine if a non vertical line is formed.
If all the points fall on the same line then you have a linear function.
First degree equations in two variables are linear functions. A first degree equation has no term with an
exponent greater than 1. Not all functions are linear.
f(x) = x2 – 3 is not a linear function because it is not a first degree equation. Remember that f(x) is
equal to the y value. We can check to see that this is true by making a function table then drawing a
graph.
x
y
-1
1
-2
-2
0
-3
1
-2
2
1
The ratio of the change in y to the corresponding change in x is called the slope of the line and is
represented by the letter (m). The slope measures the steepness of the line. If the line slants up from
left to right then the slope of the line is positive. If the line slants down from left to right then the slope
is negative. If you have a horizontal line then the slope is 0. If the line is vertical then the slope is
undefined.
To find the slope of a line on a graph you are measuring the rise of the graph against the run of the
graph. That is the vertical change in relation to the horizontal change. To determine the slope you start
at one point on the graph (it does not matter what point you start at) and you count vertically to a
second on the line (this is the change in y). Then from that point you count horizontally to to the
second point on the line for find the change in x.
slope = change in y / change in x = vertical change / horizontal change = rise/run
Chapter 5.3 Equations in Slope-Intercept form
The slope intercept form of a linear equation is: y = mx + b where m is the slope of the line and b is
the y-intercept of the line.
We know that the slope indicates the steepness and slant of the line. The y-intercept is the point where
the line crosses the y-axis. The x value for the y-intercept is always zero because the x value on the yaxis is zero.
Given one point and the slope we can graph any line. If we place the point for the y-intercept on our
graph then we use our slope to plot additional points we can graph the equation of a line in slopeintercept form.
If we have the equation y = 1/2x – 1 we can identify the slope, identify the y-intercept and graph the
line.
The slope is the m term in y = mx + b so it is the coefficient of the x term. In our example the slope of
the line is ½. That means that from any point on the graph to the next point you will go up one and
over two to find the next point. The y-intercept of a graph gives us our starting point. The y-intercept
is the constant in our equation, in the is example it is -1. So the graph will cross the y-axis at (0,-1).
If we plot (0, -1) and from this point we go up 1 unit and over 2 units we will get to our next point on
the line. We can then draw a line through the two points.
Not all linear equations are given in slope intercept form. You may need to rewrite the equation in
order to indetify your slope and y-intercept.
For example: 2y + 2x = 5 is not in slope intercept form. To place it in the form y = mx + b we first
need to get y alone on one side of the equation.
2y + 2x = 5
-2x -2x subtraction property of equality
----------------2y = -2x + 5
2
2 division property of equality
y = -x + 5/2 which is now in slope intercept form
The slope of the line (m) = -1, the y-intercept of the line(b) = 5/2
A negative slope tells us that our line will slope down from left to right. When we graph our line we
want to make sure we check the reasonableness of our line.
What do we know about the graph of y = 5/3x (this is a direct variation it is in the form y = kx. It will
pass through the origin so the y-intercept will be (0,0)
How about the line y = -5 (this is a horizontal line that crosses the y-axis at (0,5). For every x value the
y-value will be 5. The slope of the line is 0 because the coefficient of the x term is 0.
Given the slope and the y-intercept we can write the equation of a line in slope intercept form. If we
know that the slope of a line is -2 and the y-intercept of the line is -5. We can write the slope intercept
form of the line as y = -2x – 5.
You can also write an equation of a line in slope-intercept form given the graph of a line.
The y-intercept is 5 so we know that in our equation b = 5
From the y-intercept we count down 2 and over to the right 3 in order to get our slope. The slope of
our line will be -2/3 so the equation of our line is y = -2/3x + 5.
Lastly we can write the equation of a line in slope intercept form given the slope and any point on the
line. For example: The slope of the line is -5/4 and the line passes through point (-8,0)
We can use the slope intercept form of the line to solve this by using the given slope and x and y
values. 0 = -5/4(-8) + b
0 = 10 + b
-10 -10
-------------------------10 = b
The slope intercept form of the line is: y = -5/4x – 10
We can also use the formula for finding the slope to write the equation of a line in slope-intercept form.
m = y2 – y1 / x2 – x1
m = -5/4 = y- 0 / x - (-8)
-5/4 = y – 0 / x + 8
if we cross multiply we get
-5(x+8) = 4(y-0)
-5x – 40 = 4y
using the distributive property
4
4
using the division property of equality
-5/4 x – 10 = y
Chapter 5.4 Equations in point-slope form
The point slope form of a non-vertical linear equation with slope m and through point (x1, y1) is:
y – y1 = m(x – x1).
When you know the slope of a line and a point that the line contains use the point slope form to write
an equation of a line. You need to remember that if you know any two points on the line you know the
slope of the line because you can use the formula for the slope of a line ( m = y2 – y1 / x2 – x1) to
calculate the slope of the line.
For example: Given the slope m = -2 and a the line passes through point (2, -1.5) find the equation of
the line and graph it.
y-(-1.5) = -2(x-2)
y + 1.5 = -2x +4 This is the point slope form of the line. To graph this line you need to plot point
(2, -1.5) and use the slope m = -2 to place other points on the line.
-1.5
-1.5
y = -2x + 2.5
This is the same line in slope-intercept form
Our goal is to write equations of lines. Whether the line is in point slope form or slope intercept form
does not matter – it is the same line when we graph it.
You can also use the point slope form to write the equation of a line when you know two points on the
line.
You do this by finding the slope of the line given the two points using the formula for the slope
( m = y2 – y1 / x2 – x1).
You then substitute the slope and the coordinates of one of the points into the formula
y – y1 = m(x – x1)
Simplify if necessary
Example: Write the equation of a line in point-slope form passing through points (-1, 5) and (3, -5)
First we find the slope: (-5-5)/(3- - 1) = -10/4 = -5/2
Then we plug in one point and use our slope to write our equation in point slope form:
y – 5 = -5/2(x - (-1))
y – 5 = -5/2(x + 1) This is point slope form of the line
It is important to remember that the slope can be found using any two points on the line and that the
slope and either of the two points can be substituted into the point-slope form to write the equation.
Chapter 5.5 – Change the Form of a Linear Equation
The standard form of a linear equation is AX + BY = C Where A, B and C are real numbers and A and
B are both non-zero.
The standard form of a linear equation allows you to quickly find the x and y intercepts of the line. The
x cooridnate of the point where the graph crosses the x-axis is the x-intercept.
To find the x intercept, substitute 0 for y and solve for x
To find the y intercept, substitute 0 for x and solve for y
Example: Find the x and y intercepts of the equation 8x + 6y = 1200
To find the x-intercept: 8x + 6(0) = 1200, 8x = 1200, x = 150 The x intercept is (150, 0)
To find the y-intercept: 6(0) + 6y = 1200, 6y = 1200, y = 200 The y-intercept is (0, 200)
You can graph the equation equation by plotting the intercepts and and drawing a line through them.
When a linear equation is written in standard form, A or B can be zero, but not both. If either A or B is
zero, the line is horizontal or vertical.
You can transform linear equations from one form to another. You can write a linear equation in slopeintercept form, point slope form and standard form.
Equation Form
y= mx + b
m = the slope of the line
b = the y-intercept
y – y1 = m(x – x1)
m = slope
(x1,y1) = any point on the line
Example
When to Use
y = -⅔ x + 4
Use when you know the slope
and the y-intercept.
y – 6 = ⅔(x + 3)
Use when you know a point and
the slope or when you are only
given two points.
AX + BY = C
A, B and C are all real numbers, 2x + 2y = 12
and A and B are not both zero.
Use to quickly find the x and y
intercepts.
An equation written in one form can be rewritten in other forms.
Let's rewrite 5x + 2y = 20 in slope intercept form. It is currently in what form?
All points on the graph of a linear equation are solutions to the equation. Each point on the line is an
ordered pair that makes the equation true. To determine if a point is on the graph of a line, substitute
its coordinates into the equation of the line. If the result is a true statement then the point is on the line.
Example: is the point (-3, 0) on the graph 4x – 3y = -9?
4(-3) – 3(0) = -9
-12 = -9
(-3,0) is not on the graph 4x – 3y = 9
You can also determine the coordinate of a point on a line if you know one coordinate of the point and
the equation of the line.
Example: Find the value of y so that the given point lies on the given line.
(-4, y); y = -3/2x - 7 Use substitution.
y = -3/2(-4) - 7
y=6–7
y = -1
The point (-4, -1) lies on the line y = -3/2 x – 7
Chapter 5.6 Parallel and Perpendicular lines
Before we begin speaking about parallel and perpendicular lines we need to review some symbolism.
When we see AB with ↔ above it we are talking about line AB containing points A and B.
Parallel lines are lines that are in the same plane, but have no point in common, meaning that they will
never intersect.
The two lines in the above graph are parallel because the have the same slope but cross the y-axis at
different points. The both have a slope of m = 1. The two lines are part of a family of graphs because
they have at least one characteristic in common, that is, they both have the same slope.
Parallel lines:
• If two different non-vertical lines have the same slope, then they are parallel
• If two different non-vertical lines are parallel then they have the same slope
• Any two different vertical lines are parallel
To determine whether lines are parallel:
• Write each equation in slope intercept form
• Compare the slopes of the lines. If the slopes are the same and the y-intercepts are different, the
lines are parallel.
Since the slopes of parallel lines are the same, you can use this fact to write the equation of a line
parallel to a given line:
Find the slope of the line
Substitute that information into either the slope intercept form or the point-slope form of a linear
equation to write the equation of a parallel line.
When the y-intercept is given use the slope intercept form of a line to find a parallel line to a given line.
When a point is given use the point slope form to find a parallel line to a given line.
Examples 1:
Determine whether the lines are parallel:
line EF is 6x – 2y = 4 the equation of line GH is y = -3x + 8.
Rewrite EF in slope intercept form:
6x – 2y = 4
-6x
-6x
-2y = -6x + 4
-2
-2
EF in slope intercept form is : y = 3x – 2
GH in slope intercept form is: y = -3x + 8
-3 ≠ 3 The lines do not have the same slope so they are not parallel lines.
Example 2:
Determine whether the lines are parallel (||)
Given J(-4,-2), K(-4,1), L(2,-1), and M(2,5), is JK || LM?
Slope of JK = 1 – 4
-4-(-4)
= -3
0 Slope is undefined so you have a vertical line
Slope of LM = 5 - (-1)
2–2
=6
0 Slope is undefined so you have a vertical line
The lines are parallel because any two vertical lines are parallel
Write an equation of a line parallel to the graph of y = -1/3x + 2 and passing through the point (3,2).
You are given a point so you want to use the point slope form of a line.
The slope is -1/3
The point slope form of a line is y-y1 = m(x-x1)
y-2 = -1/3(x-3) Substitute the given point into the point slope form of a line
y-2 = -1/3x + 1 Simplify using the distributive property
+2
+2 Addition property of equality to simplify
y = -1/3x + 3
So y = -1/3x + 3 is parallel to y = -1/3x + 2
Perpendicular lines are lines that intersect to form right angles. The slopes of perpendicular lines are
negative reciprocals of each other and their products is -1.
The following lines are perpendicular:
The slopes of the lines are 1 and -1 respectively so the are negative reciprocals of each other and their
product is -1 (1*-1 = -1).
Perpendicular ( ┴) lines:
• If the product of the slopes of two lines is -1, then the lines are perpendicular.
• If the two lines are perpendicular and neither one is vertical, the product of their slope is -1.
• Vertical lines are perpendicular to horizontal lines.
To determine if two lines are perpendicular:
• Find the slope of each line that is given.
• Compare the slopes of the lines. If the slopes are negative reciprocals, the lines are
perpendicular.
There are many lines perpendicular to a given line. For each of them you can write an equation using
the negative reciprocal of the slope of the given line.
• Use the slope intercept form of the given line to find its slope. Find its negarive reciprocal for
the slope of the perpendicular line.
• Use this slope and the other information about the perpendicular line to write the equation.
Example 1:
Determine if the given lines are perpendicular:
Line ST is y = 5x – 1; Line UV is 10x + 2y = 12. Is ST
┴
UV?
The slope of ST = 5
The negative reciprocal of 5 is -1/5.
Put UV in slope intercept form: 10x + 2y = 12
-10x
-10x
2y = -10x + 12
2
2
y = -5x + 6
The slope of UV is -5, the slope of ST is 5, they are not negative reciprocals of each other so they are
not perpendicular.
Example 2:
l and l are
l is 4, the slope of l is x-6 / 16. Find the value of x.
┴, the slope of 1
2
Since l and l are
l
1
2
┴ the slope of 2 is -1/4 because perpendicular lines have slopes that are negative
reciprocals of each other.
1
2
x-6/16 = - ¼
Multiply both sides of the equation by 16.
16 (x-6/16) = -1/4 (16) Multiplication property of equality
x – 6 = -4
+6 +6 Addition property of equality
x=2
Example 3:
Write and equation in slope intercept form of the likne that is perpendicular to the graph of y = x + 2
and passes through the point (-1,-4).
You are given a line and a point so you are going to want to use the point slope form of a line to solve
the equation.
The slope of y = x + 2 is going to be 1. So the slope of the line perpendicular to this line is going to be
-1.
y-y1 = m(x – x1) Point slope form of a line
y-(-4) = -1(x-(-1))
y+4 = -1(x+1)
y+4 = -x – 1
-4
-4
y = -x – 5 is perpendicular to the line y = x + 2.
Chapter 5.7 Graph a linear inequality in a coordinate plane
The number sentence y ≥ - x + 4 is a linear inequality in two variables. The related equation, y = -x + 4
separates the graph into three sets of points, the points on the line, the points above the line and the
points below the line. The regions above and below the line are called half planes, the line is the
boundary of each half plane.
A solution to a linear inequality in two variables is any ordered pair that makes the inequality true. To
identify the specific solutions, substitute any ordered pair located in the identified region or on the
boundary line into the original linear inequality.
To graph the solution to a linear inequality:
•
Write the inequality in slope intercept form.
•
Graph the boundary line by replacing the inequality sign with an equal sign. If the inequality is
≤ or ≥, the boundary is a solid line, which shows that the ponts on the line are included in the
solution region. However if, the inequality is < or >, the boundary line is a dashed line, which
shows that the points on the line are not part of the solution region.
•
Shade the appropriate half plane.
•
Choose test points on the line, below the line and above the line to verify that the shaded area
represents the solution.
Example: Graph the solution to y < 2x + 3.
First we want to graph the linear equation : y = 2x + 3
Since the linear inequality contained the symbol ≤ the boundary line will be a solid line and the points
on the line will be included in the solution region.I've already graphed the "or equal to" part (it's just the
line); now I'm ready to do the "y less than" part. In other words, this is where I need to shade one side of the line
or the other. Now think about it: If I need y LESS THAN the line, do I want ABOVE the line, or BELOW?
Naturally, I want below the line. So I shade it in:
To test some points I will choose a point on the line (0,3), a point below the line (1,2) and a point above
the line (2,1).
y < 2x + 3
y < 2x + 3
y < 2x + 3
3 < 2(0) + 3
2 ≤ 2(1) + 3
1 ≤ 2(2) + 3
3<3
2≤2+3
1≤4+3
2≤5
1≤7
True the boundary
True points below the boundary
False, points above the
line is in the solution
line are in the solution.
Boundary line are not
in the solution.
Chapter 5.8 Absolute Value Functions
A function rule that contains an absolute value expression is called an absolute value function. The
graphs of absolute value functions are not linear, although they may be related to linear functions.
To graph the absolute value function y = |x|, make a function table, and graph the points.
x
-2
-1
0
1
2
y = |x|
2
1
0
1
2
The graph is V shaped and is separated into two congruent parts by a line called the axis of symmetry.
The left and the right parts of the graph are the graphs of y = -x for y ≤ 0 and y = x for y ≥ 0. The
point at which the two graphs meet is called the vertex.
An absolute value graph shows:
The vertex
The axis of symmetry
The x and y intercepts
The domain
The range
In our example:
The vertex is (0,0)
The axis of symmetry is the y-axis (x=0)
The x-intercept (0,0),The y-intercept (0,0)
The domain is the set of all real numbers
The range is y ≥ 0
An absolute value graph is a function
The graph of y = |x| is a function because
because for every x there is only one
y value.
for every x value there is only one y-value.
The graphs of absolute value functions can open up or down.
The vertex of an absolute value function is where the absolute value function equals 0.
To graph an absolute value function:
•
•
•
Make a function table.
Graph the points on the coordinate plane.
Identify the vertex (hint: the vertex is the point where the absolute value function equals 0), the
axis of symmetry the x and y intercepts, the domain, the range and determine whether the graph
opens up or down.