Chapters

Domain
Symmetry
x-intercepts
y-intercept
Asymptotes
Increase and decrease
Minimum
Maximum
Concavity and convexity
Inflection points
Graphing

Graph:

$f(x) = { x }^{ 4 } - 2 { x }^{ 2 } - 8$

Domain

$D = \mathbb{R}$

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Symmetry

$f(-x) = { (-x) }^{ 4 } - 2 { (-x) }^{ 2 } - 8 = f(x)$

Even function. Symmetry about the vertical axis.

x-intercepts

${ x }^{ 4 } - 2 { x }^{ 2 } - 8 = 0 \qquad x = \pm 2$

$(-2, 0) \qquad (2, 0)$

y-intercept

$(0, -8)$

Asymptotes

No asymptotes.

Increase and decrease

$f'(x) = 4 { x }^{ 3 } - 4x \qquad 4 { x }^{ 3 } - 4x = 0 \qquad x = 0 \quad x = \pm 1$

Increasing: $(-1, 0)U(1, \infty)$

Decreasing: $(- \infty, -1)U(0, 1)$

Minimum

$Minimum (-1, -9) \qquad Minimum (1, -9)$

Maximum

$Maximum (0, -8)$

Concavity and convexity

$f''(x) = 12 { x }^{ 2 } - 4 \qquad 12 { x }^{ 2 } - 4 = 0 \qquad x = \pm \frac { 1 }{ \sqrt { 3 } } = \frac { \sqrt { 3 } }{ 3 }$

Concave: $( - \infty, \frac { - \sqrt { 3 } }{ 3 } )U( \frac { \sqrt { 3 } }{ 3 }, \infty)$

Convex: $(\frac { - \sqrt { 3 } }{ 3 }, \frac { \sqrt { 3 } }{ 3 })$

Inflection points

Inflection points: $(-\frac {\sqrt { 3 } }{ 3 }, - \frac { 77 }{ 9 }) \qquad (\frac { \sqrt { 3 } }{ 3 }, - \frac { 77 }{ 9 })$

Graphing

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Hamza

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Graphing polynomial function II

Domain

Symmetry

x-intercepts

y-intercept

Asymptotes

Increase and decrease

Minimum

Maximum

Concavity and convexity

Inflection points

Graphing

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing rational function IV

Graphing Functions

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Concave and Convex Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Geometric Sequence Problems

Piecewise Functions

Trigonometric Functions

Arithmetic Sequence Problems

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Graphing exponential function II

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Geometric Sequence

x- and y-Intercepts

Bounded Sequence

Graphing rational function 8

L’Hôpital’s Rule

Graphing polynomial function I

Graphing exponential function I

Absolute and Relative Maxima and Minima

Maxima and Minima

Types of Functions

Asymptotes

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Inverse Function

Graphing rational function II

Graphing logarithmic function

Quadratic Function

Absolute Value Function

Rational Functions

Sequences

Bounded Functions

Logarithmic Functions

Properties of Limits

Formulas

Sequence Formulas

Exercises

Inflection Points Worksheet

Maxima and Minima Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Linear Function Word Problems