During the fourth quarter, I do a "pre-Algebra 2" unit on function families. One of the results I would like my students to see is that any function they will encounter in Algebra 2 can be written in a form similar to the "slope" form I use with my Geometry students.
When we study similar triangles, we go back and visit the equation of a line that they learned in Algebra 1 and look at it through slope eyes.
Consider a line through the points (x1,y1) and (x2,y2). Place an arbitrary point (x,y) anywhere on the line. I know that the slope triangles shown in the graph are similar, hence, the ratios of the corresponding sides are equal. In other words, the slopes as measured by these two point are equal.
Being this is the case, we can write an equation stating the two slopes are equal
The remarkable thing is we can write the equation of an absolute value graph using a similar form. If the vertex of the graph is at (x1,y1) and (x2,y2) is another known point on the graph, the equation of the absolute value graph can be written in the form
Likewise, a quadratic with a vertex at (x1,y1) and (x2,y2) another known point on the graph can be written as
Similarly, the equation of cubic (the transformed parent function y = x3 ) can be written as
Only slight modifications are needed to apply this cubic equation form to any cubic function, or to an exponential function, or even to a periodic function.
In essence, the point-slope form of a line from Algebra 1 can be modified to fit three other function families: Absolute Value, Quadratic, and Cubic.
Try using these equation forms in the applet below. Use point A for the vertex and point B for the other point on the graph. Calculate the rise and the run as you would for a linear equation. For calculating convention, your rise should calculated using yB - yA, provided point A is your vertex. Enter your equation in the input line. Click on the RESET ICON in the upper left corner of the applet to create a new pair of points.
