**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**March 2009 Goodie of the Month** – Tangents with Quadratics

Well now that we have saved all that time not factoring and completing the square and considering complex roots, we might want to study tangents to quadratics.

Now you might ask why? Answer: it reinforces so many topics from Algebra 1 and ties them all together.

How to study tangents to quadratics?

Tell the student that the slope of the tangent line to the quadratic function y(x)=ax²+bx+c is always s(x)=2ax+b.

Now ask them "Given y(x)=x²+x-3, find and graph the equation of the tangent line to this quadratic at x=-2?

1. First they must graph the quadratic by finding the y-intercept at c=-3, roots at -2.3 and 1.3 and vertex at (-0.5,-3.25).

2. Then, they must find the point on the quadratic by substituting y(-2) and check that this point (-2,-1) is on the quadratic.

3. Then, they must find the equation of the slope at this point: s(x)=2x+1.

4. Then they must find the slope of the tangent by substituting s(-2) to get m=-3.

5. Then they must find the equation of the tangent line using the point-slope formula: y= s(-2) (x+2)+y(-1).

6. Finally, they must graph this line y=-3x-7 and see that it is indeed tangent to the quadratic at the point.

So why is this a “Goodie of the Month”?

Just look at all the skills it reinforces from Algebra 1! And everything is visible and checkable!

**Related topics: **

Past: March 2009 Baddie of the month – Teaching Complex Numbers with the Quadratic Formula

Past: February 2009 Goodie of the month – Good questions for Quadratic Equations/Functions

Past: February 2009 Baddie of the month – Teaching completing the square to graph a quadratic.

Past: January 2009 Baddie of the Month – Hand-factoring a quadratic with a≠1.

I really like this GeoGebra applet and I agree that the visual nature of the whole thing makes it much more compelling. It’s interesting and satisfying to see so many concepts all tied together from a single application. My initial concern was that the student may query the significance of s(x) and hence some of the impact dissipated, but I thought that this can be overcome by using the general high school expression for slope (i.e. change in y over change in x). Then by discussing what is happening to the change in x as the secant approaches the tangent, the student should be able to come to some conclusion (maybe even with a slight addition to the applet) and actually show the expression for s(x) is correct … and a nice little introduction to limits without them even knowing.

Comment by The Math Maker — 27 March 2009 @ 7:05 pm

Absolutely cool idea about using the slope of the secant to get to s(x)! Just stating the formula for s(x) was indeed a weak point. (I actually use this topic in a very abbreviated calculus course and they take the derivative.) I will think about how to add this to the applet. Many thanks for the comment!

Comment by admin — 27 March 2009 @ 10:37 pm

There is a definite sense of limit that gets played with here – I am uneasy with it as an Algebra I topic.

If they cannot derive 2a + b, then they have to accept it, which is problematic.

Comment by Jonathan — 12 April 2009 @ 3:35 pm