College Algebra Quick Reference Sheet

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College
Course:
College Algebra
Academic year:

2023
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College Algebra Quick Reference Sheet Function Notation y = f(x) Domain Range Set Notation Set of all valid x Set of all valid y Lines or Linear Functions Slope of Line through points (x1, y1) & (x2, y2) Interval Notation Set-Builder Notation (a, b) [a, b] [a, b) {x|a 0 (positive h) or right if h < 0 (negative h). 2. Divide the x-coordinates of the points on the graph obtained in Step 1 by b. This results in a horizontal scaling, but may also include a reflection about the y-axis if b < 0 (negative b). Vertex Form The graph has a smile if a is positive and a frown if a is negative, and has a vertex at (h, k). Vertical Line through point (a, 0) Special Factoring Formulas Average Rate of Change The average rate of change m for function y=f(x) between x=a and x=b is Absolute Value Properties Special Product Formulas 3. Multiply the y-coordinates of the points on the graph obtained in Step 2 by a. This results in a vertical scaling, but may also include a reflection about the x-axis if a < 0 (negative a). 4. Add k to each of the y-coordinates of the points on the graph obtained in Step 3. This results in a vertical shift up if k > 0 (positive k) or down if k < 0 (negative k). Absolute Value Function as a Piecewise-Defined Function Quadratic Formula Solve Intercepts of an Equation x-intercepts y-intercepts Set y = 0; solve for x Set x = 0; solve for y Properties of Equality Absolute Value Equations and Inequalities If c is a positive number: Symmetry of the Graph of an Equation Type Mathematical x-axis Unchanged when y replaced by -y y-axis Unchanged when x replaced by -x origin Unchanged when y replaced by –y & x replaced by -x Geometrical Unchanged when reflected about x-axis Unchanged when reflected about y-axis Unchanged when rotated 180° about origin Properties of Inequalities If , then 2 real unequal solutions If , then 2 real duplicate solutions If , then no real solutions Factored Form for real factors: Page 1 College Algebra Quick Reference Sheet Composition of Functions Logarithm Definition Definition: End Behavior of a Polynomial Function n a Logarithm Example Behavior odd Steps for Finding the Inverse Function 1. 2. 3. 4. even even Replace with y. Interchange x and y. Solve for y. Replace y with . th n term: th n partial sum: Inverse Function Let f be a one-to-one function with domain A and range B. Then its inverse function f -1 has domain B and range A. Each point with coordinates (a, b) in f has a corresponding point (b, a) in f -1. odd Arithmetic Sequence Special Logarithms Common Logarithm Natural Logarithm where Geometric Sequence Definition: th n term: th n partial sum: Logarithm Properties Inverse Function Property Multiplicities of Real Zeros of a Polynomial Function m Behavior odd Crosses the x-axis even Touches the x-axis Let f be a one-to-one function with domain A and range B. The inverse function f -1 satisfies the following cancelation properties. Finance Formulas Laws of Logarithms Radical Properties Product Rule Quotient Rule Rational Functions For all formulas: Af is the future amount Ap is the present amount t is the number of years r is the annual interest rate (decimal) n is the number of periods in a year i = r/n is the interest rate per period R is the periodic payment amount Simple Interest Power Rule Vertical Asymptotes (No Holes) If a factor (x-a) appears in the denominator (but not in the numerator), the line x=a is a vertical asymptote. Logarithm Change of Base Formula Compound Interest Horizontal Asymptote If the degree of the numerator is less than the degree of the denominator, then there is a horizontal asymptote at y = 0 (x-axis). If the degree of the numerator is the same as the degree of the denominator, then there is a horizontal asymptote at y= (leading coefficient of numerator) / (leading coefficient of denominator). Exponent Laws and Properties Steps to Solve an Exponential Equation Continuously Compounded Interest 1. Isolate the exponential function. 2. Take the appropriate logarithm of both sides. Future Value of an Annuity 3. Use the inverse function property. 4. Solve for the variable. If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote. Present Value of an Annuity Steps to Solve a Logarithmic Equation 1. Isolate the logarithmic function. 2. Use the appropriate base to raise both sides. 3. Use the inverse function property. Payment Amount of a Loan 4. Solve for the variable. 5. Remove false answers (look for domain errors). Page 2