Math 41 Study Guide

1.1 Basic Equations

• Fractional Equations:

o Find numbers that would make the denominator equal to 0

o Multiply by LCD

• Using Radicals : Solution of is

o if n is even

o if n is odd

1.2 Modeling with Equations

• Read the problem several times

• Figure out what you have to find

• Define variable (s)

• Write down all the information, sketch a diagram if necessary

1.3 Quadratic Equations

• Factoring

• Completing the square
• Quadratic Formula :

• Discrimant:

o if D > 0 then the equation has two real solutions

o if D = 0 then the equation has one real solution (of multiplicity 2)

o if D < 0 then the equation has two complex solutions

1.4 Complex Numbers

• Addition/ Subtraction : (a + bi) ± (c + di) = (a ± c) + (b ± d)i

• Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i

• Conjugate of a + bi: a − bi
• Division: Multiply by conjugate:

• Square roots of negative numbers:

1.5 Other Types of Equations

• Polynomial Equations

• Factoring by grouping

• Equations with Radicals: Isolate the root on one side, then square/ cube /etc.

• Substitution

1.6 Inequalities

• Make the right side equal to 0

• Find the critical points: where the left side is either 0 or undefined

• Put the critical points on the number line

• Use test points

• If multiplying/ dividing by a negative number reverse the direction of the inequality

1.7 Absolute Value Equations and Inequalities

• Equations:

o Isolate the absolute value on one side

o |x| = C implies x = ±C

• Inequalities:

o Isolate the absolute value on one side

o |x| < C implies −C < x < C

o |x| > C implies x > C or x < −C

2.1 The Coordinate Plane

• Distance formula:

• Midpoint formula:
 

2.2 Graphs of Equations in two Variables

• Finding intercepts:

x-intercepts: set y equal to 0

y-intercepts: set x equal to 0

• Equation of a circle:

• Symmetry:

o x-axis: equation unchanged if y is replaced by −y

o y-axis: equation unchanged if x is replaced by −x

o origin: equation unchanged if x is replaces by −x and y by −y

2.4 Lines

• Slope:

• Slope of a vertical line is undefined, slope of a horizontal line is 0

• Point-Slope Form:

• Slope- Intercept Form : y = mx + b

• Vertical line: x = a, Horizontal line: y = b

• Parallel lines: same slope

• Perpendicula lines:

2.5 Modeling: Variation

• Direct variation: y = kx

• Inverse variation:

• Joint variation: Multiple instance of the two kinds above

3.1 What is a function

• Domain: all possible x-values. Finding the domain: Find potential problems that
could make the function undefinend such as

o Roots (even power): radicand cannot be negative

o Fractions: denominator cannot be 0

• Range: all possible y-values: Finding the range: Graph the function

3.2 Graphs of Functions

• Piece-wise defined functions

• Vertical line test: if a vertical line crosses two points of the graph, then the graph is
not a function

• Test equations if they define a function:

o Solve for y

o Check if y is unique

• Memorize the graphs of the functions on page 226

3.3 Increasing and Decreasing Functions, Average Rate of Change

• Increasing and Decreasing Functions

• Average rate of change:

3.4 Transformations of Functions

• Vertical shift: f(x) + c

• Horizontal shift: f(x − c)

• Reflection in x-axis: −f(x)

• Reflection in y-axis: f(−x)

• Vertical stretch(c > 1)/shrink(c < 1) by factor c: cf(x)

• Horizontal stretch(c < 1)/shrink(c > 1) by factor 1/c: f(cx)

• Even function: symmetric with respect to y-axis: f(x) = f(−x)

• Odd function: symmetric with respect to origin: −f(x) = f(−x)

3.5 Quadratic Functions, Maxima and Minima

• f(x) = ax² + bx + c

• Maximum if a < 0, Minimum if a > 0

• x-value of Max/Min is

3.6 Combining Functions

• Addition/Subtraction/Multiplication/Division of functions: Add/Subtract/etc. the
y-values

• Composition of functions: (f o g)(x) = f(g(x))

• Domains of combined functions

3.7 One-to-one Functions and their Inverses

• One-to-one functions: no two different x -values have the same y-value

• Horizontal line test: if a horizontal line crosses two points of the graph, the function
is not one-to-one

• Inverse function:

• Finding inverses:

o Reverse x and y

o Solve for y

• Properties: