Today we talked about how to graph functions. We started with these examples where factoring lets you find the roots (also known as zeros or x-intercepts).
\(y = 4-x^2\).
\(f(x) = x^3 - 3x^2 - 18 x\).
\(y = 2x - 6\).
\(g(x) = x^4 - 3x^3 + 2 x^2\).
We also reviewed the following six basic graphs that you should know:
We finished by using these 6 basic graphs to graph variations like:
\(y = 3|x|\)
\(y = \dfrac{1}{x-2}\)
\(y = x^2 + 5\)
\(y = |x-3|\)
\(y = -x^2\)
\(y = \sqrt{-x}\)
Today we focused on linear functions. You need to know these formulas for linear functions:
You also need to understand slope very well:
We did the following exercises in class:
Find a formula to convert Celsius to Fahrenheit.
Find a formula to convert Fahrenheit to Celsius.
Find an equation for the line passing through \((-1,4)\) and \((3,-4)\).
Find the slope and \(y\) intercept of the equation \(6x − 5y + 15 = 0\).
Find a formula for a line with slope 2 passing through \((2,5)\).
\(\dfrac{x+3}{4} = 3\)
\(\dfrac{3}{x} - 5 = \dfrac{2}{x}\)
Pressure underwater (measured in ATMs) is a linear function of the depth \(x\) in meters given by \(P = \frac{1}{10}x+1\). What is the slope and what are its units?
In 2021, the Virginia state income tax for people making over $17,000 is \(T = 720 + 0.0575 (I - 17000)\). Graph this linear function. What is the slope? What does the input variable \(I\) represent? What about the output variable \(T\)?
Today we talked about powers and radicals. We covered the basic rules for powers and radicals, and some of the common mistakes that people make. Here is a quick summary of what you should know.
We did the following exercises in class:
Simplify the following.
\(8^{2/3}\)
\(4^{5/2}\)
\(16^{1/4} + 16^{-1/4}\)
\(\sqrt{x^2+x^2+x^2+x^2}\)
\(\displaystyle \frac{m^2 n^3}{a^2 c^{-3} } \cdot \frac{a^{-7} n^{-2}}{m^2 c^4}\)
\(\displaystyle \sqrt{\frac{18x}{25x^3}}\)
Solve.
\(\dfrac{2^{57}}{2^x} = 8\)
\((x^{-1} + 2^{-1})^{-1} = \tfrac{1}{3}\)