Math 2 Semester Exam Review
Direct Proportionality (U1.L1&2 p. 7)
𝑦 = 𝑘𝑥 𝑟
r
Y varies directly with x with a constant of proportionality
of k.
Inverse Proportionality (U1.L1&2, p. 7)
𝑘
𝑦= 𝑟
𝑥
Y varies inversely with xr with a constant of proportionality
of k.
For each of the following functions, indicate whether it is an example of direct variation, inverse variation, or neither.
For those functions that are examples of direct or inverse variation, identify the constant of proportionality and write a
sentence describing the relationship.
Sketch graphs for direct variation power functions a.
when r is a positive even integer and k is positive.
b. when r is a positive odd integer and k is positive.
c. when r is a positive even integer and k is negative.
d. when r is a positive odd integer and k is negative.
Sketch graphs for inverse variation power functions
b. when r is a positive odd integer and k is positive.
a. when r is a positive even integer and k is positive.
c. when r is a positive even integer and k is negative.
d. when r is a positive odd integer and k is negative.
Consider relations expressed in forms like 𝑧 =
𝑘𝑥
𝑦
where x,y,
and k˃0.
a. How does z change as x increases?
b. How does z change a y increases?
c. What equivalent form express x as a function of y
and z?
d. What equivalent forms express y as a function of x
and z?
Consider systems of linear equations like ax+by=c and
dx+ey=f.
a. What are the possible numbers of solutions for
such systems? How are those possibilities
illustrated in graphs of the solutions for the
separate equations?
b. What methods can be used to find the solutions?
If A=[
3 7
5 −5
] and B=[
] how would you compute
2 10
11 0
A+B?
a. Write a general description of matrix addition.
What must be true about the sizes of matrices M and N in
order to compute M+N?
Consider equations expressed in the form ax+by=c.
a. How many pairs of (x,y) values can be found to
satisfy such equations?
b. What pattern can be expected in graphs of the
solution pairs?
3 4 2
], what is the size of matrix A?
9 6 5
What number is in row 1, column 2?
If A=[
−3 7 2
3 −5
] and B= [
]
1 8 4
2 0
a. Is it possible to multiply AxB?
b. Generally what must be true about the size of 2
matrices for matrix multiplication to be
possible?
c. What will be the size of the product matrix?
d. What is the product of AxB?
e. Show the steps required to calculate the entry in
row 2, column 2 of the product matrix.
If A=[
If A=[
−3 7
1
2
] how would you compute 3A?
1 8 15
a. Write a general description for how to multiply a
matrix M by a number k. (Scalar multiplication.)
How are the sizes of the matrices M and kM
related?
Commutative Property of Addition. (U2.L3 p. 134)
3 4 2
1 9 8
Suppose A=[
] and B=[
]. Is it true that
1 0 9
6 5 4
A+B=B+A?
Additive Identity (U2.L3 P. 134)
4 2
Suppose A=[−5 5]. Find a matrix C so that A+C=A.
8 7
Additive Inverse. (U2.L3 P. 134)
2 4 −3
Let a=[
]. Find the additive inverse C so that
3 −5 −7
A+C=0.
Commutative Property of Multiplication (U2.L3 P. 135)
3 7
5 −5
Suppose A=[
] and B=[
].
2 10
11 0
Is A x B = B x A?
Multiplicative Identity (U2.L3 p. 136)
3 7
If A=[
] for what matrix I, will A x I = A. Show that I
2 10
x A = A.
Multiplicative Inverse (U2.L3 p. 137)
Use your calculator to find A-1, where
−8 −10
𝐴=[
].
2
3
Show that A-1 x A = A x A-1 = I
Matrix Method to solve systems of equations (U2.L3 p.
141)
Write a matrix equation to represent the following linear
system.
X+y=5000
8x+12y = 4200
Use inverse matrices to solve for x and y.