Day 92: Logistic Function Logic
We started our study of exponential and logarithmic functions yesterday. Because much of the book work is a repeat of last year’s experience, I like to add depth and breadth to their experiences this year. Thus, we study the logistic function. As an introduction beyond seeing the equation and graph in Chapter 1, I use the Logistic Function Exploration by Paul Foerster. (This link will get you to the worksheet indirectly.)
Today, students came in with their attempts. Although #2 didn’t ask for the exponential equation for the first 6 points, we used it to review solving a system of exponential equations…easy-peezy. Then we tackled #3…lots of groans and exclaimed, “that problem was impossible,” and “I couldn’t do it.” So I played “Vanna” and the kiddos talked me through the process; used color to help connect the process some. When we used the natural log to solve for “b,” we talked at the “why” behind the canceling; great review of log and exponent properties!! One student asked if they could use the logistic form that used the denominator of 1 + aBx instead of 1 + ae-bx; this gave a great opportunity to talk about the fact that B = e-b .
Also, we didn’t use the calculator until the very end. Wanted to re-emphasize with them that they CAN solve any problem using algebra and algebra properties, and that the calculator helps us get a number we recognize, even though it isn’t exact. Such an important skill for calculus, don’t you think?
We also looked at #6 which gave them the initial value and the point of inflection. We drew the sketch and then found the equation for that logistic function. Also asked what other point they “know” is on the graph without knowing the equation. Then they have some practice problems for homework.