What is the Graph of $\,y = f(x)\,$?

Index card: 13ab

The lesson Graphs of Functions in the Algebra II curriculum gives a thorough introduction to graphs of functions.

For those who need only a quick review, the key concepts are repeated here. The exercises in this lesson duplicate (and then extend) those in Graphs of Functions.

• The equation ‘$\,y = f(x)\,$’ is an equation in two variables, $\,x\,$ and $\,y\,.$

  The graph of the equation $\,y = f(x)\,$ is the picture of all the points $\,(x,y)\,$ that make it true. Observe that to make this equation true, $\,y\,$ must equal $\,f(x)\,.$

  Thus, the graph of the equation $\,y = f(x)\,$ is the set of all points of the form $\,\color{green}{\bigl(x,\overbrace{f(x)}^{y}\bigr)}\,.$

• The graph of a function $\,f\,$ is the set of all its $\,\bigl(\text{input},\text{output}\bigr)\,$ pairs. Recall that when $\,x\,$ is an input, then $\,f(x)\,$ is the corresponding output.

  Thus, the graph of $\,f\,$ is the set of all points of the form $\,\color{green}{\bigl(x,f(x)\bigr)}\,.$

• Compare! The following two requests ask for exactly the same thing:

  Graph the equation $\,\color{green}{y = f(x)}\,.$
  
  Graph the function $\,\color{green}{f}\,.$

  Both are asking for the set of all points of the form $\,\bigl(x,f(x)\bigr)\,.$

Understanding the Relationship Between an Equation and its Graph

There are things that you can DO to an equation $\,y = f(x)\,$ that will change its graph.

Or, there are things that you can DO to a graph that will change its equation.

Stretching, shrinking, moving up/down left/right, reflecting about axes—they're all covered thoroughly in the next few web exercises:

• Shifting graphs up/down/left/right
• Reflecting about axes, and the absolute value transformation
• Horizontal and vertical stretching/shrinking

An understanding of these graphical transformations makes it easy to graph a wide variety of functions, by starting with a basic model and then applying a sequence of transformations to change it to the desired function.

For example, after mastering the graphical transformations, you'll be able to do the following:

• Need the graph of $\,y = -\sqrt{3x} + 2\,$? No problem!

  Start with the graph of $\,y = \sqrt{x}\,$ (one of the basic models):

  (A) Divide all the $x$-values by $\,3\,$ (a horizontal compression/shrink—the points get closer to the $y$-axis).

  (B) Then reflect about the $x$-axis.

  (C) Then move it up $\,2\,.$

  You'll understand why the order BAC would also work, but the order CAB doesn't!

• Or, suppose you have the graph (call it $\,G\,$) of a known equation $\,y = f(x)\,.$ The graph is being changed, though, and you need the corresponding new equations.

  If $\,G\,$ is shifted up $\,2\,$ units, the new equation is $\,y = f(x) + 2\,.$ (Transformations involving $\,y\,$ are intuitive.)

  If $\,G\,$ is shifted right $\,2\,$ units, the new equation is $\,y = f(x-2)\,.$ (Transformations involving $\,x\,$ are counter-intuitive.)

  If $\,G\,$ is reflected about the $y$-axis, the new equation is $\,y = f(-x)\,.$

For your convenience, all the graphical transformations are summarized in the Graphical Transformations tables below. Given any entry in a row, you should (eventually!) be able to fill in all the remaining entries in that row.

(Cases 18–24 below help you master the ideas in the Graphical Transformation tables!)

Summary: Graphical Transformations

Set-Up for the Tables:

• You're starting with the equation $\,y = f(x)\,.$ So, the ‘previous $\color{purple}{y}$-value’ (see the first column below) is $\,f(x)\,.$

  (For a mobile-friendly design, the columns turned into rows. Images of the original table formatting are given below.)

• Assume:
  $p \gt 0$    (‘$p$’ for Positive)
  $g \gt 1$    (‘$g$’ for Greater than)
• The point $\,(a,b)\,$ is a point on the graph of $\,y = f(x)\,,$ so that the equation $\,b = f(a)\,$ is true.

Image of original table formatting:

Image of original table formatting: