DO NOT RECOPY 8.1 Inverse Variation *Remember we did this after direct variation (2.2)*!!
8.1 Inverse Variation
Direct variation: y=kx; k≠0
k
Inverse variation: y=x; k≠0
Constant: “k”
DO NOT RECOPY 8.1 Inverse Variation *Remember we did this after direct variation (2.2)*!!
How to tell direct variation, indirect variation, neither given a table:
-If y increases as x increases, check for common quotients. Consistent? = direct variation
-If y decreases as x increases, check for common products. Consistent? = indirect variation
*If not consistent, then it’s NEITHER
DO NOT RECOPY 8.1 Inverse Variation *Remember we did this after direct variation (2.2)*!!
How to find k of indirect variation given x and y:
k
x
-Plug in x and y values to the y= formula
-Solve for k
k
-Plug k value back into y= x formula (keep x and y variables)
DO NOT RECOPY 8.1 Inverse Variation *Remember we did this after direct variation (2.2)*!!
How to graph:
-Make a table
-Use -10, -5, -2, -1, 0, 1, 2, 5, 10 for x
-Indirect variation problems ALWAYS have an asymptote at x=0
-Graph points and connect dots (making arrows straight up and down near the y-axis)
DO NOT RECOPY 8.1 Inverse Variation *Remember we did this after direct variation (2.2)*!!
8.2 The Reciprocal Function Family
1
Reciprocal function: f(x) = 𝑥 (where x ≠ 0)
Asymptote: (of a curve) a line such that the distance between the curve and the line approaches zero as they
tend to infinity *it cannot touch this line*
Branch: branches are lines out of the function. If parent function (positive a), then branches are in Q1 and Q3.
If reflected (negative a), branches are in Q2 and Q4.
How to graph:
1) Draw dashed (imaginary) asymptotes
2) Translate graph according to stretch/shrink/reflections *remember that the “parent function” has these values:
(1,1) and (-1, -1). Use branches to help!
How to write equation from translation:
1) Use asymptotes to identify h (x=h) and k (y=k).
2) Plug in/substitute this into general form
Examples:
Sketch the asymptotes and the graph of the function. Identify the domain and range.
1) 𝑦 =
1
+
𝑥−3
2) 𝑦 =
−2
𝑥
4
−3
Write an equation for the translation of 𝑦 =
3) x= -2 and y = 3
2
𝑥
that has the given asymptotes:
8.3 Rational Functions and Their Graphs
*DENOMINATOR CAN NEVER = 0!!*
*Remember to factor numerator AND denominator first!*
Holes: a place in the graph that is empty (set denominator = 0 to find that value)
*When you only have common factors on numerator/denominator*
Vertical Asymptote: a vertical line the graph cannot cross and goes up or down continuously near
*When you have no common factors on numerator/denominator (set denominator = 0 on non-common factors
to find the value(s))*
Horizontal Asymptote: a horizontal line the graph cannot cross and goes left or right continuously near
*If the degrees on numerator/denominator are same:
Divide the coefficient of the highest degree on top by
the coefficient of the highest degree on bottom
*If the degree on numerator is GREATER than denominator: No asymptote
*if the degree on numerator is LESS than denominator: Asymptote is y=0
Examples:
Find the vertical asymptotes and holes for the graph of each rational function:
(𝑥−3)
(𝑥−3)
1) (𝑥−3)
(2𝑥+2)
2) (𝑥 2 +5𝑥+6)
3) (𝑥 2 −1)
4)
(𝑥 2 −9)
5)
(𝑥−3)
(𝑥+2)
(𝑥 2 +4)
Find the horizontal asymptotes for the graph of each rational function:
2𝑥
1) 𝑥−3
1
𝑥−2
2) (𝑥 2 −2𝑥−3)
𝑦 = 𝑥+1 − 2
𝑥2
3) 2𝑥−5
4)
−2𝑥+6
𝑥−5
5)
𝑥−1
𝑥 2 +4𝑥+4
8.4 Rational Functions
DO NOT TRY TO DIVIDE OUT
THINGS THAT ARE ADDED
OR SUBTRACTED!!*
*When simplifying, multiplying, dividing, etc. the denominator of the ORIGINAL equation cannot equal zero.
So, anything that makes the denominator zero is a “restriction” on the variable
Multiplying Rational Expressions
Steps:
1) Write both terms as fractions
2) Multiply (squish parentheses together *do not distribute*)
3) Factor all things (top and bottom)
4) Divide out/cancel any common factors
Dividing Rational Expressions
“Keep it, Switch it, and Flip it”
Steps:
1) Keep first fraction the same
2) Switch the division sign to multiplication
3) Flip the second fraction
4) Follow steps 1-4 of multiplication
Examples: Write in simplest form. State any restrictions on the variables. (2 lines)
𝑥 2 −9
24𝑥 3 𝑦 2
49−𝑧 2
1) 2
2)
3)
2 3
𝑥 +𝑥−12
−6𝑥 𝑦
𝑧+7
Multiply or Divide. State any restrictions on the variables. (2 full lines)
4)
6)
𝑥 2 +7𝑥+10
2(𝑥 2 −4)
𝑥 2 +8𝑥+12
𝑥 2 −7𝑥+10
∙
4
(𝑥+5)2
÷
𝑥 2 +10𝑥+24
𝑥 2 +𝑥−6
5)
7)
2𝑥 4
10𝑦 −2
∙
5𝑦 3
4𝑥 3
𝑥 2 −5𝑥+6
𝑥 2 −4
÷
𝑥 2 +3𝑥+2
𝑥 2 −2𝑥−3
8.5 Adding and Subtracting Rational Expressions
Like denominators:
-Add the numerators
-Keep the denominator
-Factor & Simplify
Unlike denominators:
-Identify LCD (least common denominator)
-Multiply each expression by a "creative one" so it has the LCD
-Add or subtract fractions with LCDs
-Factor & Simplify
Simplify each sum or difference. State any restrictions on the variables.
1)
4)
Find the LCM:
2)
3)
5)
6)
7)
8)
Simplify each sum or difference. State any restrictions on the variables.
9)
10)
11)
Simplify each complex fraction.
𝟏
𝟒
12)
𝟑
𝟐−
𝟓
𝟏−
𝟏
𝟑
13) 𝟑
𝐛
14)
𝟑
𝐲+
𝟐
𝐱
8.6 Solving Rational Equations
2 Ways to Solve:
Cross Multiply (if proportion)
-solve for variable
Use LCD (if adding fractions)
-multiply both sides by LCD *should cancel denominators*
-simplify & solve for variable
**ALWAYS simplify and check your answer for "extraneous solutions";
(extraneous solutions are based off restrictions on variables)
denominator ≠0!
Examples: *Leave lots of room…* 5 full lines or so for each problem
1)
2)
3)
4)
5)
6)
7)