MM - Graphs of Functions Lesson

Graphs of Functions

Earlier in this unit, you learned how to graph some special types of functions. Now we will review the key features and transformations of the graphs of all functions. These apply to the new functions, functions learned in previous modules and functions you may learn in the future. If you do not have a graphing calculator, you can use an online one.

Transformations are how the graph is moved vertically and horizontally, stretches and shrinks, and reflections.  The following chart summarizes these transformations.

To Graph	Draw by	Change in function
Vertical Shifts
Y = f(x) + k, k > 0	Shift graph up by k units	Add k to f(x)
Y = f(x) - k, k > 0	Shift graph down by k units	Subtract k from f(x)
Horizontal shifts
Y = f(x + h), h > 0	Shift graph left by h units	Replace x with (x + h)
Y = f(x - h), h > 0	Shift graph right by h units	Replace x with (x - h)
Vertical Stretches
Y = a * f(x), a > 1	Vertical stretch, makes the graph steeper	Multiply f(x) by a constant
Y = a * f(x), 0 < a < 1	Vertical shrink, makes the graph less steep	Multiply f(x) by a constant
Horizontal Stretches
Y = f(a*x), a > 1	Horizontal shrink, makes the graph steeper	Multiply x by a constant
Y = f(a*x), 0 < a < 1	Horizontal stretch, makes the graph "less steep"	Multiply x by a constant
Reflections
Y = -f(x)	Reflects across the y-axis	Multiply f(x) by a negative
Y = f(-x)	Reflects across the x-axis	Multiply x by a negative

*For certain functions, such as quadratic and absolute value functions, vertical stretches and horizontal stretches are dependent upon what form the function is written.

Here is an example with a step function.

Parent function:  $f\left(x\right)=\left[\left[x\right]\right]$                                     

Transformed function:  $f\left(x\right)=-\frac{1}{3}\left[\left[x-2\right]\right]+1$

Transformation-                

• " - " reflects the graph across the x axis                                                                      
• "1/3" shrinks the graph vertically
• "-2" in the double brackets moves the graph 2 units to the right
• "+1" moves the graph 1 unit up

If you replace the parent function with any of the functions shown below, the transformations would be the same.  

This would also be a good time to review the shapes of the graphs of our various functions.

Let's start with different polynomial functions. See the following chart below.

Polynomial Function	Example	Degree	Leading Coefficient	Graph
Constant	$f\left(x\right)=2\:or\:f\left(x\right)=2x^0$	0	2
Linear	$f\left(x\right)=x-1$	1	1
Quadratic	$f\left(x\right)=2x^2-1$	2	2
Cubic	$f\left(x\right)=x^3-2x^2+1$	3	1
Quartic	$f\left(x\right)=x^4-2x^2+1$	4	1

Now let's look at other functions, including our new ones.

Type of function	Example	Graph
Rational	$f\left(x\right)=\frac{1}{x}$
Radical	$f\left(x\right)=\sqrt[]{x}$
Exponential	$f\left(x\right)=2^x$
Logarithmic	$f\left(x\right)=\log_{10^x}$
Piecewise	$f\left(x\right)=\begin{Bmatrix} x+1& x\le 2 \\ 2x-2&x> 2 \\ \end{Bmatrix}$
Step	$f\left(x\right)=\left[\left[x\right]\right]$
Absolute Value	$f\left(x\right)=\left|x-3\right|$

Key features of graphs include domain, range, intercepts, extrema, intervals of increasing and decreasing, intervals of positive and negative, symmetry (even and odd), end behavior, and asymptotes. You may need to go back and review some of these; they are defined in the Key Terms. Here are a few that we will specifically use.

• Rational roots are points on the x - axis where the graph touches or crosses that are rational numbers.
• Irrational roots are points on the x - axis where the graph touches or crosses that are irrational numbers.
• Non-real roots are solutions to a polynomial function that are "imaginary" numbers.
• Relative maximum points are turning points where the graph changes from increasing to decreasing.
• Relative minimum points are turning points where the graph changes from decreasing to increasing.
• End behavior is how the graph acts as it goes to negative infinity and to positive infinity.

Some things to highlight in reviewing key features.

Even, Odd, Neither?

Another key feature that hasn't been discussed yet is whether a function is even, odd, or neither.

A function is even if it has y axis symmetry and odd if it has origin symmetry.  

Recalling the definitions of y axis symmetry and origin symmetry:

• The graph is even if f(x) = f(-x).  In other words, it contain both (x,y)  and (-x,y).
• A function f(x) is even if the graph is symmetric with respect to the y-axis.
• The graph is odd if f(x) = -f(-x).  In other words, it contain both (x,y) and (-x,-y).
• A function f(x) is odd if the graph is symmetric with respect to the origin.

 A shortcut for this that only  works with polynomial functions is:                                  

• If all the powers of the variable are even, the function is even.
• If all the powers of the variable are odd, the function is odd.

But you must understand that a constant has a variable power of zero; $2=2x^0$  .   So  $y=x^3-2x+5$    is neither odd nor even. The powers are 3, 1, and 0.

End Behavior

We can also calculate the average rate of change, or the slope of a function. To do this, we find 2 points and calculate the change in y-coordinates  related to the change in x-coordinates.

If the function (given in algebraic or graph form) is not  linear, we can calculate the approximate rate of change  for a section of the graph the same way.

Rate of Change

It's now time for us to explore and practice working with Graphs of Functions.

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