1 2 Elementary Functions Graphs and Transformations In

1. 2 Elementary Functions; Graphs and Transformations • In this presentation, you will be given an equation of a function and asked to draw its graph. You should be able to state how the graph is related to a “standard” function. It is not important that you plot a great many points for each graph. It IS important that you recognize the general shape of the graph. You can verify your answers using a graphing calculator, but only after you have attempted to construct the graph by hand.

Problem 1 • Construct the graph of

Solution

Problem 2 • Now, sketch the related graph given by the equation below and explain, in words, how it is related to the first function you graphed.

Solution: Problem 2 • The graph has the same shape as the original function. The difference is that the original graph has been translated two units to the right on the x-axis. Conclusion: The graph of the function f(x-2) is the graph of f(x) shifted horizontally two units to the right on the x-axis. • Notice that replacing x by x-2 shifts the graph horizontally to the right and not the left.

Problem 3 • Now, graph the following “standard” function: Complete the table: -3 -2 -1 0 1 2 3

Solution to problem 3

Problem 4 • Now, graph the following related function:

Solution to problem 4

Problem 4 solution • The graph of • is obtained from the graph of • by translating the graph of the original function up one unit vertically on the positive y-axis.

Problem 5 • Graph: • What is the domain of this function?

Solution to problem 5 • The domain is all non-negative real numbers. Here is the graph:

Problem 6 • Graph: • Explain, in words, how it compares to problem 5.

Problem 6 solution (Notice that the graph lies entirely within the fourth quadrant)

Graph of –f(x) • The graph of the function –f(x) is a reflection of the graph of f(x) across the x-axis. That is, if the graphs of f(x) and –f(x) are folded along the xaxis, the two graphs would coincide.

Cube root function • Sketch the graph of the cube root function. Complete the table of ordered pairs: x y -27 -8 -1 0 1 8 27

Variation of cube root function • Sketch the following variation of the cube root function:

Same graph as graph of cube root function. Shifted horizontally to the left one unit.

Graph of f(x+c) compared to graph of f(x): • The graph of f(x+c) has the same shape as the graph of f(x) with the exception that the graph of f(x+c) is translated horizontally to the left c units when c >0 and is translated horizontally to the right c units when c < 0.

Absolute Value function • Now, graph the absolute value function. Be sure to choose x values that are both positive and negative as well as zero.

Graph of absolute value function Notice the symmetry of the graph.

Variation of absolute value function

Shift absolute value graph to the left one unit and down two units on the vertical axis.