Rational Equations and Functions - Andrews University
Rational Equations and Functions Algebra II Chapter 8 This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L.
2011 Holt McDougal Some examples and diagrams are taken from the textbook. Slides created by Richard Wright, Andrews Academy [email protected] 8.1 Model Inverse and Joint Variation
Direct Variation: y = ax x , y Inverse Variation: x , y Joint Variation: y = axz y depends on both x and z a is the constant of
variation 8.1 Model Inverse and Joint Variation What type of variation is each of the following? xy = 48 2y = x
y = 2x + 3 8.1 Model Inverse and Joint Variation Solving Variations Plug in x and y to find a Plug in a and the other value and solve y varies inversely as x. When x = 2, y = 6. Write an equation
relating x and y. Then find y when x = 4. 8.1 Model Inverse and Joint Variation Checking data for variation Plug each of the data points in one of the variation equations to find a If the a stays the same, the data has that type of variation What type of variation?
X 2 4 8 y 8
4 2 8.1 Model Inverse and Joint Variation Writing variations from sentences y varies directly with x and inversely with z2
z varies jointly with x2 and y y varies inversely with x and z 555 #3-33 odd, 39, 41 + 2 = 20 total Quiz
8.1 Homework Quiz 8.2 Graph Simple Rational Functions Rational Functions Functions written as a fraction with x in the denominator
Shape called hyperbola Asymptotes Horizontal: x-axis Vertical: y-axis 8.2 Graph Simple Rational Functions General form a stretches vertically (multiplies y-values) h moves right k moves up
How i transformed from ? 8.2 Graph Simple Rational Functions How to find asymptotes Vertical Make the denominator = 0 and solve for x
8.2 Graph Simple Rational Functions Horizontal Or
Substitute a very large number for x and estimate y Find the degree of numerator (N) Find the degree of denominator (D) If N < D, then y = 0 If N = D, then y = leading coefficients
If N > D, then no horizontal asymptote Find the asymptotes for 8.2 Graph Simple Rational Functions Domain All xs except for the vertical asymptotes Range
All the ys covered in the graph Usually all ys except for horizontal asymptote 8.2 Graph Simple Rational Functions Graph by finding asymptotes
and making a table Graph 561 #1, 3-31 every other odd, 39, 41 + 4 = 15 total Quiz 8.2 Homework Quiz
8.3 Graph General Rational Functions Find the asymptotes Simplify first Factor and cancel entire factors Vertical take the denominator = 0 and solve for x 8.3 Graph General Rational Functions
Substitute a very large number for x and estimate y Find the degree of numerator (N) Find the degree of denominator (D) If N < D, then y = 0 If N = D, then y = leading coefficients If N > D, then no horizontal asymptote Find the asymptotes for
8.3 Graph General Rational Functions How to find x-intercepts Let If Only happens if numerator = 0 How to find y-intercepts Let and simplify
8.3 Graph General Rational Functions To graph rational functions Find the asymptotes Make a table of values around the vertical asymptotes Graph the asymptotes and points Start near an asymptote, go through the points and end near another asymptote Each graph will have several sections NEVER cross a vertical asymptote
8.3 Graph General Rational Functions Graph 568 #3-15 odd, 19, 23, 33, 35 + 4 = 15 total
Quiz 8.3 Homework Quiz 8.4 Multiply and Divide Rational Expressions Simplified form numerator and denominator can have no common factors
Steps to simplify Factor numerator and denominator Cancel any common factors 8.4 Multiply and Divide Rational Expressions Simplify 3
2 +5 +6 3 2 +2 8.4 Multiply and Divide Rational Expressions Multiplying Rational Expressions Factor numerators and denominators
Multiply across top and bottom Cancel factors 8.4 Multiply and Divide Rational Expressions 8.4 Multiply and Divide Rational Expressions Dividing Rational Expressions
Take reciprocal of divisor Multiply 8.4 Multiply and Divide Rational Expressions Combined Operations Do the first two operations Use that result with the next operation
577 #3, 7-17 odd, 25-43 odd, 49 + 2 = 20 Quiz 8.4 Homework Quiz 8.5 Add and Subtract Rational Expressions Adding and Subtracting Need least common denominator (LCD)
If LCD already present, add or subtract numerators only To get fractions with LCD Factor all denominators LCD is the common factors times the unique factors Whatever you multiply the denominator by, multiply the numerator also 8.5 Add and Subtract Rational Expressions
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