Algebra 1 Chapter 3 Study Guide
Topic
3.1 Graphing Linear Equations
Things to know…
Standard Form Ax+By=C
Constraints
1) A ≥ 0
2) A and B are not both 0
Determining if an equation is a linear
equation
Find x-intercepts and y-intercepts
Graph by using intercepts
3.2 Solving Linear Equations by Graphing
Equation
Solution or root
Function
Zeros and x-intercepts
Solve an equation with one root
Rewriting equation as function (remember
0 must be on the right hand side!)
Graphing equations using 3 points
Estimating by graphing
3.3 Rate of Change and Slope
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = 𝑆𝑙𝑜𝑝𝑒 =
3.4 Direct Variation
3.5 Arithmetic Sequences
𝑚=
Comparing rates of change
Determining if a function is linear by
finding a constant rate of change
Positive, negative, and zero slope
(horizontal line)
Undefined slope (vertical line)
Find coordinate given slope
Direct Variation y=kx, where k≠0
Constraints
1) Contains the origin (0, 0)
2) Constant rate of change
called constant of
variation denoted k
Graph direct variation
Write and solve direct variation equation
Direct variation as a function
Estimate using direct variation
nth term Arithmetic Equation
𝑎 = 𝑎 + (𝑛 − 1)𝑑
n is the term number
d is the common difference
an is the number in the sequence
that is the nth term
Identify arithmetic sequence (is there a
common difference d? If so then yes!)
Find the next term
Find the nth term
Given a number in the sequence,
find what term it is in the
sequence.
Given a term in the sequence, find
the value of the term in the
sequence.
Write the equation for nth term of the
arithmetic sequence
There should be two variables in
the equation an and n.
Arithmetic sequences as functions
Proportional relationships
Line passes through the origin (0,0)
Ratio between x-values and yvalues is the same.
Note: proportional relationships can be written as
direct variations.
Nonproportional relationships
Ratio between x-values and yvalues is different.
Line does not pass through the
origin.
Write equations of proportional and
Nonproportional relationships.
3.6 Proportional and Nonproportional
Relationships
Examples
3.1 Graphing Linear Equations
3.2 Solving Linear Equations by Graphing
3.3 Rate of Change and Slope
Find the value of r so the line that passes through each pair of points has the given slope.
1) (3, 5), (-3, r), m = ¾
𝑚=
Write the formula for slope
( ) ( )
) ( )
=(
=
3(−6) = 4(𝑟 − 5)
−18 = 4𝑟 − 20
−18 + 20 = 4𝑟 − 20 + 20
Substitute the values into the equation
Simplify
Find the cross product
Solve for r
2 = 4𝑟
2 1
𝑟= =
4 2
𝟏
𝒓=
𝟐
3.4 Direct Variation
Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve.
1) If y = 4.5 when x = 2.5, find y when x =12.
What are we given? x=2.5 y=4.5
What do we want to find? A direct variation equation and the value of y when x is 12.
y = kx
(4.5) = k(2.5)
.
∗ .
= .
.
k=1.8
y=1.8x
Write direct variation equation
Substitute the given values for x and y into the equation
Solve for k
Substitute this value for k back into y=kx
This is the direct variation equation that relates x and y.
We now use this equation we have found to find y when x is 12.
y=1.8x
Write direct variation equation you found
y=1.8(12)
Substitute x with 12
y=21.6
21.6 is the value when x is 12
3.5 Arithmetic Sequences
Given the sequence 4, 9, 14, 19, …
a) Determine if the sequence is an arithmetic sequence.
Is there a common difference d?
9-4=5, 14-9=5, and 19-14=5. Since the difference between any two successive terms is 5, we have a
common difference of d=5.
This is an arithmetic sequence with d=5
b) Write an equation for the nth term of the arithmetic sequence.
𝑎 = 𝑎 + (𝑛 − 1)𝑑
Write the nth term equation for the arithmetic sequences
𝑎 = 4 + (𝑛 − 1)(5)
Substitute the values we know into the equation d=5 and a1=4
𝑎 = 4 + 5𝑛 − 5
Distribute the 5 to (n-1)
𝒂𝒏 = 𝟓𝒏 − 𝟏
Simplify
Note: notice that the only variables in the equation are an and n
c) Find the 9th term of the sequence.
This means n=9
Write the nth term equation you found for this sequence
Substitute 9 where there is an n in the equation
The 9th term of the sequence is 44
𝑎 = 5𝑛 − 1
𝑎 = 5(9) − 1
𝒂𝟗 = 𝟒𝟒
d) Which term of the sequence is 84?
This means that an=84
𝑎 = 5𝑛 − 1
84 = 5𝑛 − 1
84 + 1 = 5𝑛 − 1 + 1
85 = 5𝑛
=
Write the nth term equation you found for this sequence
Substitute an with 84
Solve for n
Add 1 to both sides
Divide both sides of the equation by 5
n=17
84 is the 17th term of the sequence
3.6 Proportional and Nonproportional Relationships
1) Write an equation in function notation given the table
x
Y
0
-2
2
4
4
10
6
16
Find the difference between successive x-values and the difference between successive y-values.
The difference between the x-values is 2.
The difference between the y-values is 6.
𝑠𝑙𝑜𝑝𝑒 =
= =3
Find the common difference in the x-values
Find the common difference in the y-values
Find the slope
The y intercept is -2.
f(x)=mx+b
f(x)=3x-2
Find the y-intercept
Substitute the m with the slope you found and b with
the y-intercept.
2) Is the relationship between the x and y values proportional or nonproportional.
Since the y-value is not 0 when the x-value is 0, the function does not pass through the origin.
At this point you know that the relationship in nonproportional.
Also notice that
≠ ≠
≠
.