Unit 3 of 8

Understanding Individual Graphs

  • 14 lessons
  • Parts 35 to 62

Each basic function has a graph worth knowing by heart, with its domain, range, intercepts and long run behavior. This unit finds domains and ranges, then studies quadratics, integer powers, square and cube roots, and the reciprocal functions one at a time.

Piecewise functions, polynomial zeros and sign charts, and rational functions with holes and asymptotes come next. The unit ends with shifting, reflecting, stretching and compressing graphs, so a new formula can be sketched from a familiar shape.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Domain and Range

  1. Finding the Domain

    How to find the natural domain of a real formula by listing every root and denominator condition and keeping only the inputs that satisfy all of them at once.

    Theory part 35Practice part 36

  2. Finding the Range

    How to find the range of a function on its domain by bounding the outputs and then showing, with an explicit input, that every proposed output really occurs.

    Theory part 37Practice part 38

Powers and Roots

  1. The Quadratic Function and Its Graph

    How to use the standard, vertex and factored forms of a quadratic to find its vertex, axis, intercepts, opening direction and range, and sketch the parabola from those facts.

    Theory part 39Practice part 40

  2. Cubic and Other Integer-Power Functions

    How the graphs of x squared, x cubed and higher powers get their shapes, why even and odd exponents give different symmetry, and how powers compare below and above 1.

    Theory part 41Practice part 42

  3. The Square-Root Function

    What the principal square root means, why its graph starts at an endpoint and only rises, and how to find the domain, range and exact points of a formula like the square root of 6 minus x.

    Theory part 43Practice part 44

  4. The Cube-Root Function

    Why every real number, negative ones included, has exactly one real cube root, and how to build the graph of the cube root and its shifts from exact points chosen with perfect cubes.

    Theory part 45Practice part 46

Reciprocal Functions

  1. The Reciprocal Function

    How the graph of 1 over x comes from the equation xy = 1, why it has two separate branches and no point at zero, and what its vertical and horizontal asymptotes mean.

    Theory part 47Practice part 48

  2. The Reciprocal-Square Function

    Why the graph of 1 over x squared is a pair of mirror image branches above the axis, what its domain, range and asymptotes are, and why doubling the input quarters the output.

    Theory part 49Practice part 50

Piecewise, Polynomial, and Rational Graphs

  1. Reading and Drawing Piecewise Functions

    How to read a piecewise rule by checking the input conditions first, evaluate it at any input, and draw its graph with the right open and closed endpoints, jumps and joins.

    Theory part 51Practice part 52

  2. Polynomial Zeros, Signs, and Graph Shape

    How the factors of a polynomial give its zeros and their multiplicities, how a sign chart and the leading term fix where the graph is above or below the axis, and what such a sketch can and cannot show.

    Theory part 53Practice part 54

  3. Sign Charts for Products and Quotients

    How to solve product and quotient inequalities by marking every zero and excluded input, reading signs factor by factor on each interval, and testing the endpoints on their own.

    Theory part 55Practice part 56

  4. Rational Functions, Holes, and Asymptotes

    How to find the excluded inputs of a rational function from its original denominator, and how to tell a hole from a vertical asymptote and read the far end behavior.

    Theory part 57Practice part 58

Moving and Scaling Graphs

  1. Shifting and Reflecting Graphs

    How shifts and reflections move each point of a known graph, why the input that reproduces an old output decides the horizontal move, and how to find the new domain.

    Theory part 59Practice part 60

  2. Stretching and Compressing Graphs

    How multiplying a function's output differs from multiplying its input, how each one moves the points of a graph, and how to read the scale factor in each direction.

    Theory part 61Practice part 62