Precalculus

• Functions


  One of the most important topics in precalculus
• Graphing functions


  1. How to graph functions

  2. How to find the domain and range
• In this video, the different types of functions, how the graphs look, and how to find the domain and range will be covered
• Linear function


  Graph is a straight line passing through the origin
• Linear function


  • Domain is negative infinity to infinity
  • Range is negative infinity to infinity
• Quadratic function


  Graph is a parabola
• Quadratic function


  • Domain is all real numbers
  • Range is from 0 to infinity
• Cubic function


  Graph is an increasing function
• Cubic function


  • Domain is all real numbers
  • Range is from negative infinity to infinity
• Square root function


  Graph increases at a decreasing rate
• Square root function


  • Domain is from 0 to infinity
  • Range is from 0 to infinity
• Cube root function


  Graph is symmetric about the origin
• Cube root function


  • Domain is from negative infinity to infinity
  • Range is from negative infinity to infinity
• Absolute value function


  Graph forms a V-shape and opens upward
• Absolute value function


  • Domain is all real numbers
  • Range is from 0 to infinity
• Rational function 1/x


  Graph has a horizontal asymptote at y=0 and a vertical asymptote at x=0
• Rational function 1/x


  • Domain is from negative infinity to 0 union 0 to infinity (excluding 0)
  • Range is from negative infinity to 0 union 0 to infinity (excluding 0)
• Rational function 1/x^2


  Graph is similar to 1/x but the left side flips over the x-axis
• Rational function 1/x^2


  • Domain is from negative infinity to 0 union 0 to infinity (excluding 0)
  • Range is from 0 to infinity (excluding 0)
• Exponential function e^x


  Graph has a horizontal asymptote at y=0 and increases at an increasing rate
• Exponential function e^x


  • Domain is all real numbers
  • Range is from 0 to infinity
• Natural log function ln(x)

  Graph is the inverse of the exponential function and has a vertical asymptote at x=0
• Natural log function ln(x)

  • Domain is from 0 to infinity
  • Range is from negative infinity to infinity
• Sine function


  Graph is a sinusoidal wave
• Sine function


  • Domain is from negative infinity to infinity
  • Range is from -1 to 1
• Cosine function


  Graph is a sinusoidal wave starting at the top
• Cosine function


  • Domain is from negative infinity to infinity
  • Range is from -1 to 1
• Tangent function


  Graph has vertical asymptotes at odd multiples of π/2
• Tangent function


  • Domain is all real numbers except odd multiples of π/2
  • Range is from negative infinity to infinity
• Transformations of functions


  1. Vertical stretch/shrink by a constant factor

  2. Horizontal shrink/stretch by a constant factor
• Parent function


  The original function before any transformations
• If you put a 2 in front of the parent function

  It stretches the graph vertically by a factor of 2
• If you put a 1/2 in front of the parent function

  It shrinks the graph vertically by a factor of 2
• If you put a 2 inside the parent function

  It shrinks the graph horizontally by a factor of 2
• If you put a 1/2 inside the parent function

  It stretches the graph horizontally by a factor of 2
• If you have f(x-4)

  The graph shifts 4 units to the right
• If you have f(x+3)

  The graph shifts 3 units to the left
• If you put a negative on the outside of the function


  It reflects the graph over the x-axis
• If you put a negative on the inside of the function


  It reflects the graph over the y-axis
• If you put a negative on the outside and inside of the function


  It reflects the graph over the origin