Today we talked about quadratic polynomials and we did this Workshop on Parabolas.
We started by looking at two functions that have vertical asymptotes.
\(f(x) = \dfrac{x+6}{x-3}\)
\(y = \dfrac{1}{x^2-4}\)
Any time you get a \(\dfrac{\text{nonzero}}{\text{zero}}\) when you plug an \(x\)-value into a function, that \(x\)-value will be a vertical asymptote.
Today we introduced the idea of continuity.
Intuitively a function is continuous if you can draw the graph without lifting your pencil.
Functions constructed from addition, multiplication, powers, and composition of continuous functions are always continuous everywhere they are defined.
The only places where the y-values of a function can cross from positive to negative are the roots or the discontinuities (i.e., bad points where the function is not defined).
You can use the last idea to solve inequalities. We did the following examples.
Solve \(\frac{(x-3)(x+1)}{(x-1)^2} > 0\)
Solve \(x+3 > \frac{4}{x}\)
Solve \(|2x-7| > 3\)
Solve \(|2x - 7| \le 3\).
I recommend the following strategy to solve inequalities.
Replace the inequality by an equality and find the \(x\)-values where the two sides are equal. These are the crossing points.
Find any bad points where one side is not defined (or discontinuous).
Draw a number line and mark all of the crossing points and bad points. These marks cut the number line into several subintervals.
Test one representative point from each subinterval to see if the inequality is true or false there.
Here is a good video explanation of this technique.