Solve for x
1. 15 + loge (27x − 3) = 4
2. 7(5x ) = 4(52x )
3. loge (4x 2 − 2x ) = loge (2x ) − 37
4. 2(e x ) 5x + 8 = 9
2
Graphing rational functions
r (x ) =
8(x − 3)
−5(x + 1)(x 2 − x + 4)
1. What are the vertical asymptotes?
2. What are the x-intercepts
3. Is r (x ) positive or negative between pairs of points from 1 & 2?
4. What’s the leading term of 8(x − 3)? Of −5(x + 1)(x 2 − x + 4)? What’s the quotient of those
leading terms?
5. What does the graph of r (x ) look like on the right and left?
6. Graph r (x )
Page 2
Graphing exponential and logarithmic functions
1. e x
2. loge (x )
3. e x−3
4. −2 loge (x )
Page 3
Graphing piecewise defined functions

x + 2
1. f (x ) = 
 (x − 1) 2



|x + 3|



2. д(x ) = 
4



5 − x



2x − 2




3. h(x ) =  −1


 −4x + 16

if x ∈ (−∞, 1)
if x ∈ [1, ∞)
if x ∈ (−∞, −3]
if x ∈ (−3, 2]
if x ∈ (2, ∞)
if x ∈ (−∞, 0)
if x ∈ [0, 3)
if x ∈ [3, ∞)
Page 4
Absolute value inequalities
1. Solve |2x + 3| < 6
2. Solve |π − 3x | < 2
Page 5
Linear equations in two and three variables
1. How many solutions does a linear equation in two variables have? (e.g. ax + by = r such
that at least one of a and b is nonzero) How many solutions does a linear equation in three
variables have?
2. There are three possibilities for the number of solutions of a system of linear equations.
What are they?
3. When does a system of two linear equations in two variables have a unique solution?
ax + by = r
cx + dy = s
4. Is x = −1 and y = −5 a solution of the following system?
−3x + y = 2
x +y = 6
Page 6
Solve for x and y
1. Solve
3x + 2y = 19
x +y = 8
2. Solve
4x + 3y = −2
8x − 2y = 12
Page 7