Week 2 Review

Definitions

100

What is natural log equivalent to?

Log base e.

200

What is a domain?

The set of all possible input values (usually represented by x). Essentially, it is everything you can "plug into" an equation or graph and get a valid, real output.

300

What is an Inverse (or tell us how to find an inverse function).

A function that serves to "undo" another function. You can find it by switching x and y, then solving for y.

400

When is a slant asymptote formed?

When the degree of the numerator is exactly one more than the degree of the denominator.

500

What is the product property of Log?

log_bm*n = log_bm + log_bn.

Rational Functions

100

If the degree of the numerator is ______ than the degree of the denominator, then you will have a horizontal asymptote at y=___

<, 0

200

The degree of the numerator is ___ the degree of the denominator when there is not a horizontal asymptote when graphing a rational function.

>

300

In the reduced rational function, describe the process for determining holes in the graph.  

Factor the numerator and denominator. If there are factors (x-n) that cancel in the numerator and denominator, then the function has a hole at x=n

400

If the degree of the numerator is = to the degree of the denominator, then you will have a horizontal asymptote at y= ______ / _______

The leading coefficient of the numerator over the leading coefficient of the denominator (ratio of coefficients).

500

You _____ cross a horizontal asymptote. You ____ cross a vertical asymptote.

can, cannot

You Solve

100

(x² − 6x + 19) ÷ (x − 1)

x - 5 + (14/x-1)

200

(4x²+3x-8) ÷ (x+2)

4x-5+ (2/x+2)

300

(2x⁵+5x⁴-2x³+2x²-2x+3) ÷ (x+3)

2x⁴-x³+x²-x+1

400

(4x²+x³+2x+5) ÷ (x-2)

x²-2x-2+ (1/x-2)

500

(3x³+4x+11) ÷ (x²-3x+2)

3x+9 + (25x-7/x²-3x+2)

You Graph

100

g(x)=ln(x)-4

Slide 1

200

y=(x-1)/(x²-x-6)

Slide 2

300

g(x)=log(x+5)

Slide 3

400

(x²-2x-3) / (x²+5x+4)

Slide 4

500

(2x²-12x+16) / (x²-x-12)

Slide 5

*Extra* Problems

100

Let f(x) = 7^x and g(x) = -7^(x+4)-2. Describe the transformations, ordered correctly, from f(x) to g(x). 

Reflection over the x-axis, left 4, down 2.

200

Identify all asymptotes of f(x) = (x+4)/(x²-x-20).

HA: y = 0 

VA: x = 5

300

How can you check if 2 functions, f(x) and g(x), are inverses of each other?

If f(g(x)) = x and g(f(x)) = x, then they are inverses.

400

A biology lab starts a controlled experiment with a Petri dish containing 27 active bacteria cells. Under ideal conditions, the bacteria population quadruples every hour. Write a function B(t) that represents the total number of bacteria cells present after t hours.

B(t) = 27 ⋅ 4^t

500

Is x+4 a factor of x² + 2x + 1?

No, because there is a remainder of 9 when we do (x² + 2x + 1)/(x+4).