Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 5: Basic functions and transformations

Topic. This lesson covers Session 5: Basic functions and transformations

Learning Outcomes.

  • Translate between geometric transformations (shifting, stretching, flipping) in either direction (vertically, horizontally) and the corresponding algebraic transformations of a function
  • Identify even and odd symmetries.

WeBWorK. There are two WeBWorK assignments on today’s material:

  1. Functions – Translations
  2. Functions – Symmetries

Additional Video Resources.

Question of the Day: When you flip a graph over, what happens to the formula?

Transformations of graphs

Shifting a graph up or down: Add or subtract a number c to the output f(x), so y=f(x)+ c.

  • Consider the graph of a function y=f(x). Then the graph of y=f(x)+c is that of y=f(x) shifted up or down by c. If c is positive, the graph is shifted up, if c is negative, the graph is shifted down.

Shifting a graph left or right: Add or subtract a number c to the input x, so y=f(x+c)

  • Consider the graph of a function y=f(x). Then the graph of y=f(x+c) is that of y=f(x) shifted left or right by c. Careful: If c is positive, the graph is shifted left, if c is negative, the graph is shifted right.

Stretching or compressing a graph vertically: Multiply the output f(x) by a positive number c, so y=c\cdot f(x)

  • Consider the graph of a function y=f(x) and let c>0. Then the graph of y=c\cdot f(x) is that of y=f(x) stretched away or compressed towards the x-axis by a factor c. If c>1, the graph is stretched away from the x-axis, if 0<c<1 then the graph is compressed towards the x-axis.

Stretching or compressing a graph horizontally: Multiply the input x by a positive number c, so y=f(c\cdot x)

  • Consider the graph of a function y=f(x) and let c>0. Then the graph of y=f(c\cdot x) is that of y=f(x) stretched away or compressed towards the y-axis by a factor c. If c>1, the graph is stretched away from the y-axis, if 0<c<1 then the graph is compressed towards the y-axis.

Reflect a graph horizontally or vertically:

  • To reflect vertically (across the x-axis), multiply the output of the function by -1, so y=- f(x).
  • To reflect horizontally (across the y-axis), multiply the input of the function by -1, so y=f(-x).

VIDEO: Transformations of functions

Video by Irania Vazquez

Symmetries: Odd and Even functions

Definition. A function f is called even if f(-x)=f(x) for all x.
Similarly, a function f is called odd if f(-x)=-f(x) for all x

Observation.

  • An even function is symmetric with respect to the y-axis (if you reflect the graph horizontally across the y-axis, you end up with the same graph – the left side is a mirror image of the right side).
  • An odd function is symmetric with respect to the origin (if you rotate the graph 180^\circ about the origin, you end up with the same graph).

VIDEO: Odd and Even Functions

Video by Irania Vazquez

Exit Question

Is the function shown below odd or even?

Graph of function similar to bell curve.

Answer

The function is even, since the right side looks like a mirror image of the left side.

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.