“Is the square root of -1 equal to just 1?”

There’s a way to check that. Or any similar problem with a square root.

Take what you think is the answer, 1 and square it. If you get -1, then yes, it is. If not, it isn’t.

1(1) = 1

Not -1, so no it is not.

The solution is an imaginary number. Using both real numbers and imaginary numbers is called ‘complex’.

The square root of -1 is called i. So i squared is -1.

Matrix Addition, Subtraction, Multiplication – Tutoring Math

We worked on matrices.

Starting with matrix addition and subtraction, which were not a problem.

Then got into multiplication  by numbers as well as addition and subtraction. Sometimes factoring was helpful there.

For matrix multiplication, you multiply the row elements by the column elements and add them up to get the products elements. Sometimes if the dimensions of the matrices do not match up correctly, you cannot multiply them.

Compound interest and half lives – tutoring precalculus

We looked at problems with continuous interest in terms of banking and similar problems for elements with radioactive decay. The same equation can basically be used for both situations, you can change letters if it seems to make more sense that way.

The lnx and e^x functions can counteract each other, much like arcsinx and sinx.

There is another equation for half life specifically, but it’s easy enough to get to an equation with that idea using the original equation and therefore not memorizing more equations than necessary.

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.