Algebra II Notes – Unit Nine: Rational Equations and Functions
Syllabus Objectives: 9.1 – The student will solve a problem by applying inverse and joint variation. 9.6 – The student will develop mathematical models involving rational expressions to solve real-world problems.
Recall – Direct Variation: y varies directly with x if , where k is the constant of variation and .
Inverse Variation: x and y show inverse variation if where k is the constant of variation and .
Ex: Do the following show direct variation, inverse variation, or neither?
a. Solve for y.
b. Solve for y.
c. Solve for y.
Ex: x and y vary inversely, and when . Write an equation that relates x and y. Then find y when .
Use the equation . Substitute the values for x and y.
Solve for k.
Write the equation that relates x and y.
Use the equation to find y when .
Application Problem Involving Inverse Variation
Ex: The volume of gas in a container varies inversely with the amount of pressure. A gas has volume 75 in.3 at a pressure of 25 lb/in.2. Write a model relating volume and pressure.
Use the equation . Substitute the values of V and P.
Solve for k.
Write the equation that relates V and P.
Checking Data for Inverse Variation
W / 2 / 4 / 6 / 8 / 10H / 9 / 4.5 / 3 / 2.25 / 1.8
Ex: Do these data show inverse variation? If so, find a model.
If , then there exists a constant, k, such that .
Check the products from the table.
The product k is constant, so H and W vary inversely.
Joint Variation: when a quantity varies directly as the product of two or more other quantities
Ex: The variable y varies jointly with x and z. Use the given values to write an equation relating x, y, and z.
Use the equation . Substitute the values in for x, y, and z.
Solve for k.
Write the equation that relates x, y, and z.
Ex: Write an equation for the following.
1. z varies jointly with and y.
2. y varies inversely with x and z.
3. y varies directly with and inversely with z.
You Try: The ideal gas law states that the volume V (in liters) varies directly with the number of molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in kilopascals). The constant of variation is denoted by R and is called the universal gas constant.
Write an equation for the ideal gas law. Then estimate the universal gas constant if V = 251.6 liters; n = 1 mole; t = 288 K; P = 9.5 kilopascals.
QOD: Suppose x varies inversely with y and y varies inversely with z. How does x vary with z? Justify your answer algebraically.
Sample CCSD Common Exam Practice Question(s):
The value of s varies jointly with m and p. If s = 10 when m = 3 and p = 4, what is the value of s when m = 7 and p = 2?
A.
B.
C.
D.
Syllabus Objectives: 9.2 – The student will graph rational functions with and without technology. 9.3 – The student will identify domain, range, and asymptotes of rational functions. 9.6 – The student will develop mathematical models involving rational expressions to solve real-world problems.
Rational Function: a function of the form , where and are polynomial functions
x / −4 / −3 / −2 / −1 / / / 1 / 2 / 3 / 4y / / / / −1 / −2 / 2 / 1 / / /
Ex: Use a table of values to graph the function .
Note: The graph of has two branches. The x-axis is a horizontal asymptote, and the y-axis is a vertical asymptote. Domain and Range: All real numbers not equal to zero.
Exploration: Graph each of the functions on the graphing calculator. Describe how the graph compares to the graph of . Include horizontal and vertical asymptotes and domain and range in your answer.
1.
2.
3.
4.
5.
6.
Hyperbola: the graph of the function
Ex: From the exploration above, describe the asymptotes, domain and range, and the effects of a on the general equation of a hyperbola .
Horizontal Asymptote: Vertical Asymptote:
Domain: All real numbers not equal to h. Range: All real numbers not equal to k.
As gets bigger, the branches move farther away from the origin. If , the branches are in the first and third quadrants. If , the branches are in the second and fourth quadrants.
Ex: Sketch the graph of
Horizontal Asymptote: Vertical Asymptote:
Plot points to the left and right of the vertical asymptote:
Sketch the branches in the second and fourth quadrants.
More Hyperbolas: graphing in the form
Horizontal Asymptote: Vertical Asymptote:
Ex: Sketch the graph of .
Horizontal Asymptote: Vertical Asymptote:
Plot points to the left and right of the vertical asymptote:
Application Problems with Rational Functions
Ex: The senior class is sponsoring a dinner. The cost of catering the dinner is $9.95 per person plus an $18 delivery charge. Write a model that gives the average cost per person. Graph the model and use it to estimate the number of people needed to lower the cost to $11 per person. What happens to the average cost per person as the number increases?
Model: Average cost = (Total Cost) / (Number of People)
They need at least 17 people to lower the cost to $11 per person.
The average cost approaches $9.95 as the number of people increases.
You Try: Write a rational function whose graph is a hyperbola that has a vertical asymptote at and a horizontal asymptote at . Can you write more than one function with the same asymptotes?
QOD: In what line(s) is the graph of symmetric? What does this symmetry tell you about the inverse of this function?
Sample CCSD Common Exam Practice Question(s):
What is the graph of ?
Syllabus Objectives: 9.2 – The student will graph rational functions with and without technology. 9.3 – The student will identify domain, range, and asymptotes of rational functions. 9.6 – The student will develop mathematical models involving rational expressions to solve real-world problems.
Graphs of Rational Functions
· x-intercepts: the zeros of
· Vertical Asymptotes: occur at the zeros of
· Horizontal Asymptote: describes the END BEHAVIOR of the graph (as )
o If the degree of is less than the degree of , then is a horizontal asymptote.
o If the degree of is equal to the degree of , then y = (the ratio of the leading coefficients) is a horizontal asymptote.
o If the degree of is greater than the degree of , then the graph has no horizontal asymptote.
Ex: Graph the function .
· x-intercepts:
· Vertical Asymptotes:
· Horizontal Asymptote:
· Plot points between and outside the vertical asymptotes
x / −2 / / 0 / / 2y / / / 0 / /
Note: The point lies on a horizontal asymptote. This is valid since horizontal asymptotes describe end behavior. A graph can NEVER have a point exist on a vertical asymptote since vertical asymptotes represent where the function does not exist.
Local (Relative) Extrema: the local (relative) maximum is the largest value of the function in a local area, and the local (relative) minimum is the smallest value of the function in a local area
Ex: Graph the function . Find any local extrema.
· x-intercepts:
· Vertical Asymptotes:
· Horizontal Asymptote:
· Plot points between and outside the vertical asymptotes
x / −4 / / 0 / 1 / 4y / 4 / / 0 / / 4
· Local maximum is 0. It occurs at the point .
Slant Asymptote: If the degree of the numerator is one greater than the degree of the denominator, then the slant asymptote is the quotient of the two polynomial functions (without the remainder).
Ex: Sketch the graph of . Use the graphing calculator to check your answer and to find the local extrema.
· x-intercepts:
· Vertical Asymptotes:
· Horizontal Asymptote: none
· Slant Asymptote:
· Plot points to the left and right of the vertical asymptote
x / −9 / / / 0 / 2 / 4y / −19.2 / / 2.5 / / / 0.63
Local Minimum: −0.8345 Local Maximum: −19.165
Application Problems with Local Extrema
Ex: A closed silo is to be built in the shape of a cylinder with a volume of 100,000 cubic feet. Find the dimensions of the silo that use the least amount of material.
Volume of a Cylinder:
Using the least amount of material is finding the minimum surface area, S, of the cylinder.
Surface Area of a Cylinder:
Substitute h from above:
Graph the function for surface area and find the minimum value.
The minimum surface area occurs when the radius is 25.15 ft. The height is
You Try: Sketch the graph of .
QOD: Describe how to find the horizontal, vertical, and slant asymptotes of a rational function.
Sample CCSD Common Exam Practice Question(s):
What are the asymptotes of the function ?
A. y = 0, x = –5
B. y = 0, x = –5, x = 5
C. y = –1, x = –5
D. y = –1, x = –5, x = 5
Syllabus Objective: 9.4 – The student will simplify, add, subtract, multiply, and divide rational expressions.
Recall: When simplifying fractions, we divide out any common factors in the numerator and denominator
Ex: Simplify .
The numerator and denominator have a common factor of 4. They can be rewritten.
Now we can divide out the common factor of 4. The remaining numerator and denominator have no common factors (other than 1), so the fraction is now simplified.
Simplified Form of a Rational Expression: a rational expression in which the numerator and denominator have no common factors other than 1
Simplifying a Rational Expression
1. Factor the numerator and denominator
2. Divide out any common factors
Ex: Simplify the expression .
Factor.
Divide out common factors.
Recall: When multiplying fractions, simplify any common factors in the numerators and denominators, then multiply the numerators and multiply the denominators.
Ex: Multiply . Divide out common factors.
Multiplying Rational Expressions
1. Factor numerators and denominators (if necessary).
2. Divide out common factors.
3. Multiply numerators and denominators.
Ex: Multiply .
Factor.
Divide out common factors.
Multiply.
Ex: Find the product.
Factor.
Divide out common factors.
Multiplying Rational Expressions with Monomials
Use the properties of exponents to multiply numerators and denominators, then divide.
Ex: Multiply .
Use the properties of exponents and simplify.
Recall: When dividing fractions, multiply by the reciprocal.
Ex: Find the quotient.
Dividing Rational Expressions
Multiply the first expression by the reciprocal of the second expression and simplify.
Ex: Divide.
Multiply by the reciprocal.
Factor and simplify.
Ex: Find the quotient of and .
Multiply by the reciprocal.
Factor and simplify.
You Try: Simplify.
QOD: What is the factoring pattern for a sum/difference of two cubes?
Sample CCSD Common Exam Practice Question(s):
Simplify the expression:
A. 1
B.
C.
D.
Syllabus Objective: 9.4 – The student will simplify, add, subtract, multiply, and divide rational expressions.
Recall: To add or subtract fractions with like denominators, add or subtract the numerators and keep the common denominator.
Ex: Find the difference.
Adding and Subtracting Rational Expressions with Like Denominators
Add or subtract the numerators. Keep the common denominator. Simplify the sum or difference.
Ex: Subtract.
Ex: Add.
Recall: To add or subtract fractions with unlike denominators, find the least common denominator (LCD) and rewrite each fraction with the common denominator. Then add or subtract the numerators.
Ex: Add.
Note: To find the LCD, it is helpful to write the denominators in factored form.
Adding and Subtracting Rational Expressions with Unlike Denominators
1. Find the least common denominator (in factored form).
2. Rewrite each fraction with the common denominator.
3. Add or subtract the numerators, keep the common denominator, and simplify.
Ex: Find the sum.
Factor and find the LCD. LCD =
Rewrite each fraction with the LCD.
Add the fractions.
Note: Our answer cannot be simplified because the numerator cannot be factored. We will leave the denominator in factored form.
Ex: Subtract:
Factor and find the LCD. LCD =
Rewrite each fraction with the LCD.
Subtract the fractions.
Complex Fraction: a fraction that contains a fraction in its numerator and/or denominator
Simplifying a Complex Fraction
Method 1: Multiply every fraction by the lowest common denominator.
Method 2: Add or subtract fractions in the numerator/denominator, then multiply by the reciprocal of the fraction in the denominator.
Ex: Use Method 1 to simplify the complex fraction.
Multiply every fraction by the LCD = and divide out the common factors.
Simplify the remaining fraction.
Ex: Use Method 2 to simplify the complex fraction.