Pre-Calculus Essentials in 3 Minutes

Transcript

Pre-calculus is the toolbox you carry into calculus, and finals tend to sample from four big drawers: trigonometry and the unit circle, exponentials and logarithms, functions and their transformations, and sequences. Let's take them one at a time.

Start with the unit circle. It's a circle of radius one, and for any angle, the cosine is the x coordinate and the sine is the y coordinate of the point where the angle's ray meets the circle. Memorize the key values. At an angle of pi over six, sine is one-half. At pi over four, sine and cosine are both root two over two. At pi over three, sine is root three over two. When you solve an equation like two times sine of x equals one, divide both sides by two to get sine of x equals one-half, then ask: where on the circle is the y coordinate one-half? At pi over six and at five pi over six.

Next, exponentials and logarithms. A logarithm is just an exponent question in disguise. Log base two of x equals five is asking: two raised to what power gives x? So x equals two to the fifth, which is thirty-two. Going the other way, log base three of eighty-one equals four, because three to the fourth power is eighty-one. If you can translate between the log form and the exponent form, most log problems solve themselves.

Third, function composition and transformations. Composition means feeding one function into another, working inside out. If f of x is x squared and g of x is x plus three, then f of g of two means: first compute g of two, which is five, then compute f of five, which is twenty-five. For transformations, remember that adding outside the function shifts the graph up, and replacing x with x minus a number shifts it to the right.

Finally, sequences. An arithmetic sequence adds the same amount each step; the nth term equals the first term plus the quantity n minus one times the common difference. Take the sequence four, nine, fourteen, nineteen. The difference is five, so the tenth term is four plus nine times five, which is forty-nine. A geometric sequence multiplies by the same ratio instead.

The classic mistake: evaluating a composition in the wrong order. F of g of x means g happens first — inside out, always.

Recap: read sine and cosine off the unit circle, rewrite logs as exponents, compose functions from the inside out, and build sequence terms from the first term plus the pattern. Test yourself with the quiz below.