Which Letters Of The Alphabet Are Graphs of Polynomials?

This light exercise is to get your students thinking about polynomial graph behavior. You could extend this exercise to guess they degrees and signs of the leading coefficients if desired. Simply print out a worksheet of all the letters of the alphabet and ask your students which letters are graphs of polynomials based on the following distinctions

1. Polynomial graphs are continuous. You can draw them without lifting your pen
2. Polynomial graphs have no sharp corners or cusps, they are smooth (see pic below).
   
   
   
   
   
   

(From www.mathisfun.com)

See mathisfun.com for a great tutorial and pics.

Tips For Teaching Horizontal & Vertical Shifts

To help your students understand vertical and horizontal shifts in graphs they need to start thinking in terms of x/y-intercepts, not x/y values. We will label this the method the horizontal slide vs the vertical slide. 

Horizontal vs Vertical Slides of Function Graphs

Here is the graph of f(x) = x^2

Here is the graph of x^2 with a Vertical shift of 2 units (f(x) = x^2 + 2)

Here is a graph of x^2 with a horizontal shift of 2 units f(x) = (x-2)^2

Students' Trouble In Understanding

Most students tend to understand vertical shifts. It seems intuitive to them that adding 2 to x^2 will shift the graph 2 units in the positive direction. However, students tend not to understand the horizontal shifts. It seems backwards to them. The reason for this is that students are concentrating on what is being done to the variables as opposed to the x/y-intercepts. The task of this activity is not mastery but to shift the students' focus to what's happening with the intercepts instead of the what is being done to the variable.

What You Will Need

• graph paper 
• multiple color markers
• Activity Sheet

Steps:

Step 1: On blank (x,y) coordinate graphing paper have students plot the following graph by generating random points.

f(x) = x^2

Step 2: Have students analyze the graph and determine the x-intercept and the y-intercept.

Step 3: Ask them what would you need to do to the graph of x^2 to change the y-intercept.

Step 4: Have them redraw the graph of x^2 anywhere else they want on the y-axis as long at it doesn't shift to the left or right.

Step 5: Ask the question how many units did your graph shift upward or downward?

Step 6: Have them contemplate what their new function will look like. Will it be x^2 plus 2, minus 2, multiplied by 2, divided by 2, etc. 

Step 7: Show them what the new function will look like

f(x) = x^2 + 2

Step 8: Have them determine what the function would like if their graph what shifted up 2 more units. What would it look like if it was shifted down 5 units?

Step 9: Have them draw the two new graphs and write the new functions beside them.

Step 10: Ask them what changed in the graph, what remained the same.

Repeat the process with Horizontal shifts, having them concentrate on the x-intercept as opposed to what is being done to the variable x

Graphs of Functions In Nature

For Algebra teachers we know there are certain "go-to" graphs of functions that tend to be most popular in our texts. Having the ability to recognize these graphs enables the student to anticipate the shape of the graph and any of its variations. Simply knowing its normal shape allows students to visualize any transitions or reflections in the graph.

The point of this exercise is to send your students on a scavenger hunt in nature for the graphs of functions. Such an exercise would make a great recess or study hall period activity as well as an end of the class exercise.  

Another Variation of Plotting Points and Graphing Lines

Instead of plotting points the normal way here is a variation

Educator may choose to replace the nails with tac's and the board with cardboard or other products like construction paper. The purpose is, as always, to let the students engage the material.

The same concepts may be used for more advanced plotting also

y = x

y= x^2 (hard to hold the string and take pictures while watching your 2-year-old :)

What else could you do? 

Matching Lines To Equations

Does the above look familiar? You give your students an equation, ask them to pick arbitrary points for x in order to solve for y. Then take those points, plot them,  and graph the line. Here's a variation from paper and pencil.

Instead of plotting point, try matching lines with equations  

Bolting two pieces of wood together we create an X, Y coordinate. 

Using a straight stick, the students goal is to hold the stick in a way that represents the line drawn when the points were plotted.