Algebra 2 Honors - Unit 4

4.1 Inverse Variation and the Reciprocal Function

k = constant of variation

Direct Variation: as one variable grows (x), the other grows (y)

Formula: $y=kx$ , “y varies directly with x”

Inverse Variation: as one variable (x) grows, the other shrinks (y)

Formula: $y=\frac{k}{x}$ , “y varies inversely with x”

Joint Variation: if y varies jointly as x and z

Formula: $y=kxz$ , “y varies jointly with x and z”

Direct and Inverse Variation:

Formula: $z=\frac{ky}{x}$ , “z varies directly with y and inversely with x”

KNOW THE RECIPROCAL FUNCTION (y=1/x) GRAPH

As $x\to\infty,\frac{1}{x}\to0$

As $y\to-\infty,\frac{1}{x}\to0$

As $x\to0^{+},\frac{1}{x}\to\infty$

As $x\to0^{-},\frac{1}{x}\to-\infty$

It gets close to 0 but will never be zero, and it’ll never cross or touch the x-axis and y-axis

4.2 - Graphing Rational Functions 1

Fractions/Rational numbers: ratios of integers

Rational Functions: ratios of polynomial functions

Formula: $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$

How to graph a transformed reciprocal function using the equation $y=\frac{3}{x-1}+2$

Find the vertical and horizontal asymptotes, which will become your new axes
1. Vertical = the opposite of the value with the x
  1. Here, the vertical asymptote is x=1
2. Horizontal = the added or subtracted number on the outside
  1. Here, the horizontal asymptote is y=2
3. These also serve as your domain and range
  1. So Domain = {x|x ≠ 1}
  2. Range = {y|y ≠ 2}
4. Put those as dashed lines on the graph
5. Find the x and y intercepts, and translate the starting points (1,1) and (-1,-1) based on the translation from the original axes to the new axes from the vertical and horizontal asymptotes
6. Draw the graph

What if it’s IMPROPER???

If an equation only has the x in the denominator, then it’s proper and you have all the transformations
If not, then it’s improper and you need to divide to make it proper
Ex: for $y=\frac{x-1}{x-3}$ , you need to do (x-1) divided by (x-3). Then you can do the transformation

4.2B - Graphing Rational Functions 2 (Skipped for now, come back to this)

4.3 - Multiplying and Dividing Rational Expressions

Holes: when dividing an improper equation, certain variables are removed, and they’re holes

Ex: in $y=\frac{x+1}{x^2-1}\to\frac{x+1}{\left(x-1\right)\left(x+1\right)}\to\frac{1}{x-1}$ , the (x+1) was removed. So if you plug in -1 into the new equation $y=\frac{1}{x-1}$ , you get y = -1/2, so the hole is at (-1,-1/2)

On the graph it’s just marked as an open circle

Also put it into the domain, so the domain is {x|x ≠ 1, x ≠ -1} for both the domain of $y=\frac{1}{x-1}$ and (x-1)

Oblique Asymptotes: asymptote that appears when there’s no horizontal asymptote

it’s a slanted asymptote

CHECK 4.2a page 3 for example

Multiplication: Factor and reduce first
1. Ex: $\frac{x^2+6x+8}{x^2+4x+3}\cdot\frac{x^3+1}{x+4}$
Division too

JUST PRACTICE THESE QUESTIONS ON THE NOTES

4.4 - Addition and Subtraction

Least Common Multiple: the smallest expression that each term divides into

How to find the LCM with 8, 10, and 6

Factor the numbers (2³, 2×5, 2×3)
Multiply all of the numbers that don’t repeat amongst all of them, alongside the number that does repeat (but the one with the biggest power, so here that’s 2³)
1. So 5×3×2³=120

PRACTICE THE WORKSHEET NOTES QUESTIONS

4.4b - Complex (Compound) Rational Expressions

JUST PRACTICE THE PROBLEMS ON THE WORKSHEET NOTES

4.5A - Solving Rational Equations

JUST PRACTICE THE PROBLEMS ON THE WORKSHEET NOTES

4.5b - Using Rational Equations

Work Problems: rate problems where you need to find the RATE per TIME for each worker to complete part of the job

Formula: (time it takes for the job to get done with all workers)*((rate of worker 1) + (rate of worker 2)+…+(rate of worker n)) = 1

Current Problems: If the current (wind or water) helps the rate, ADD. If not, SUBTRACT.

LOOK AT PROBLEMS ON WORKSHEET NOTES AND PRACTICE THOSE

4.5c - Formulas

This is just about converting formulas for a specific variable, can practice but not a priority