Section 11-1 CC: Simplifying Rational Expressions.
Objectives:
To simplify rational expressions.
Rational Expression: an expression with a variable in the denominator.
To Simplify a Rational Expression:
1)Factor the numerator if possible.
2)Factor the denominator if possible.
3)Reduce the fraction (cancel out).
Excluded Value: the value of a variable which makes the denominator become zero.
Section 11-2 CC: Multiplying and Dividing Rational Expressions.
Objectives:
To multiply and divide rational expressions.
To Multiply Rational Expressions:
1)Factor the numerators if possible.
2)Factor the denominators if possible.
3)Multiply straight across.
4)Reduce the fractions and Cross Cancel out.
To Divide Rational Expressions:
1)Multiply the first fraction by the reciprocal of the second fraction.
2)Factor the numerators if possible.
3)Factor the denominators if possible.
4)Multiply straight across.
5)Reduce the fractions and Cross Cancel out.
Section 11-3 CC: Dividing Polynomials.
Objectives:
To divide Polynomials using long division.
The Divisor and the Dividend must be written in standard form.
y + 5 ) 2y² + 3y - 40 y + 5 → divisor
2y² + 3y - 40 → dividend
When the dividend is in standard form and a power is missing →that power must be represented with a 0 as it’s coefficient.
Example: 4b³ + 5b – 3 → 4b³ + 0b² + 5b – 3
Section 11-4 CC: Adding and Subtracting Rational Expressions.
Objectives:
To add and subtract rational expressions.
Least Common Denominator (LCD): the smallest multiple of the denominators.
To Add & Subtract Rational Expressions:
1)Factor the denominators if possible.
2)If the denominators are the same:
A)Add or subtract the numerators.
B)Keep the same denominator.
C)Reduce the fraction.
3)If the denominators are NOT the same:
A)Find the LCD.
B)Rewrite each fraction with the LCD.
C)Add or subtract the numerators.
D)Keep the same denominator.
E)Reduce the fraction.
Section 11-5 CC: Solving Rational Equations:
Objectives:
To solve rational equations & proportions.
Proportion: two fractions that are equal to each other.
To Solve Rational Equations:
1)Factor the denominators if possible.
2)Find the LCD.
3)Multiply every term in the equation by the LCD.
4)Reduce/clear out the fractions.
5)Solve the equation.
6)Check your answers to make sure the denominators do NOT become zero.
Section 11-6 CC: Inverse Variation.
Objectives:
To solve inverse variations.
To compare direct and inverse variations.
Direct Variation: a linear function in the form of
y = kx .
A direct variation is a linear function.
The graph of a direct variation always passes through the origin.
k → constant of variation = slope.
k = y
x
Inverse Variation: an equation in the form of
xy = k
k → constant of variation
Section 11-7 CC: Graphing Rational Functions.
Objectives:
To graph rational functions.
Rational Function: a function that contains an x of degree one or higher in the denominator.
Asymptotes: horizontal and vertical lines that guide the graph of a rational function.
To find the Horizontal Asymptote:
y = the number to the right of the fraction.
Example: y = 2 + 4 y = 4 is the horizontal
x – 2 Asymptote.
To find the Vertical Asymptote:
Set the denominator equal to zero and solve.
Example: y = 2 + 4 x – 2 = 0
x – 2 x = 2 is the vertical
Asymptote.
Functions
1)Linear Functions – highest power of x is 1. They form a straight line on the graph.
The equations are in the form of y = mx + b.
The equation of a horizontal line is in the form of y = the y-intercept.
The equation a vertical is in the form of x = the x – intercept.
A VERTICAL LINE is NOT a Function.
2)Quadratic Functions – the highest power of x is 2. They form a parabola that opens upward or downward. The equations are in the form of:
y = ax²
y = ax² + c
y = ax² + bx + c
3)Absolute Value Functions - the variable “x” is contained inside the Absolute Value Symbols.
The graph forms a V – shape that opens upward or downward.
4)Exponential Functions – are in the form of y = a · bx .
The graph forms a curve upward or downward.
5)Square Root Functions – otherwise known Radical Functions – the variable “x” is contained in the radicand.
The graph forms a curve to the right.
6)Rational Functions – contains the variable “x” in the denominator.
The graph forms 2 curves and there are asymptotes that guide the graph.
To Find the Horizontal Asymptote: y = the number to the right of the fraction.
To find the Vertical Asymptote: Set the denominator equal to zero and solve the equation for x.