Tutoring Precalculus, Logarithms

We went over problems involving logarithms.

Log has a base of 10 if nothing is written and ln has a base of e. e is ~ 2.7, it’s a number with a lot of significance, kind of like pi.

log 10 = 1
log 100 = 2
You can think about it as a base, an exponent, and a solution.

e^x and lnx are inverse functions, meaning they can counteract each other, like sin (sin^-1(x))

Sometimes doing some algebra initially before jumping into the logarithms is helpful.

Looked a bit at the shape of lnx.

“How do I write the equation of a circle with center (5,-7) and radius 9 units?”

I would write this equation so it’s similar to how the equation of an ellipse would look. The denominators will be the same. And you could write this in different ways, but I think this way clearly shows the center and radius.

Having one on the right side of the equation, shows you the radius in the denominators and would show the major and minor axes if it was an ellipse.

Parabola focus and directrix, p value, tutoring precalculus

We reviewed for the test.

Started with parabolas. (h, k) stands for the vertex. The x coordinate being h and the y coordinate being k. In alphabetical order, just like x, y.

The focus is a distance p from the vertex as is the directrix, in opposite directions. The focus is inside the parabola.

If rays of light hit a parabola, they go to the focus.

Also looked at ellipses and hyperbolas.

Tutoring Precalculus, Permutations and Combinations

Started with sequences. Arithmetic and Geometric. Looked at the relevant formulas then used them.

He needed to figure out whether series were arithmetic or geometric.

And looked at convergence for geometric series.

Also looked at permutations and combinations. Permutations are like a telephone number, the order matters. With combinations, they are more like pizza toppings- mushrooms and olives is like olives and mushrooms, the order does not matter.

Many calculators can compute these.

Tutoring precalculus, asymptotes, holes and end behavior

We mostly looked at functions with polynomials that had asymptotes and holes.

One polynomial had the form of a sum of two cubes, where knowing a related formula is helpful.

To get the coordinates of holes, you plug the x-coordinate that gets a factor of zero in the denominator into the reduced equation.

Finding the behavior close to the asymptote is important, it generally goes up very high or down very low.

To find end behavior, I plug in high magnitude positive and negative numbers. The terms with the highest exponents become more important then.

Tutoring Precalculus, Hyperbolas

We mostly looked at hyperbolas.

For the equations, one term (x or y squared) is positive and the other is negative. The graph will be either vertical for positive y or horizontal for positive x.

h,k are the center. The asymptotes will go through the center.

Sometimes associating something with an image can help. I thought of b/a corresponding with the letter b and being horizontal.

For factoring cubic equations, if you have one zero, you can divide using that information and then get to a quadratic function.

Tutoring Precalculus, Separating Terms with summations as exponents

We started by looking at arithmetic and geometric sequences and sums.

Arithmetic is more like addition, geometric has more to do with multiplication.

Subscripts are used as labels. If the number is next to a letter, it looks like multiplication.

Separating a number with a summation for an exponent into two pieces can be useful.

And either ln or log is often useful for problems with exponents (especially when they are more complicated).

One test problem was worded in a strange way.

Tutoring Precalculus, Log function tricks

We started with some problems involving logarithms and natural logarithms.

By acting the log function on something with an exponent, the exponent can be brought in front.

Continuous compound interest uses a function with the natural number e to the power of the rate multiplied by time.

Also went over test corrections.

One trick was divide both sides of an equation by two to simplify everything.

Vertex form of quadratic equation, Tutoring Precalculus

We started by looking at some problems with division of polynomials. Chose to use synthetic division, long division would be an option also.

To fully factor, for example, a fourth degree polynomial, if it comes out without a remainder then the combination of all factors is the fully factored form. Like you could factor 12 as 2*3*2.

Another problem seemed to call for the vertex form of a quadratic equation y = a(x-h)^2 + k where (h, k) is the vertex. Then you could convert that to standard form.

Talked a bit more about imaginary numbers.

“exact value of cos 255 degrees without a calculator” Tutoring Precalculus

We started with some problems involving bearings. The direction can get mixed up pretty easily, but the problems were fairly straightforward.

Another problem asked for the exact value of cos 255 degrees without a calculator. It seemed like a half-angle formula might be a good approach.

One problem involved area which meant needing to know the area formula for a triangle 1/2 bh.

And using inverse trig functions was useful. Other problems in the review did require the law of cosines and the law of sines.