Graphing Hyperbolas, Directrix, Focus

We worked on some upcoming material in the class with periodic functions.

Drew an angle on the unit circle and found the tangent of that angle. You draw the angle and a right triangle then find the ratio using SOHCAHTOA.

Graphed a parabola and looked at the focus and directrix. Also talked about some of the practical purposes of the shape of the parabola. If you forget the equations for these, you can google them if you know what to call them.

Did a problem with a hyperbola. And saw how the sign in front of the variables affects the orientation of the shape.

Looked at a little physics as well. For sound and light with higher wavelengths, the wavelength is shorter.

Tutoring Physics, Light, Indices of Refraction

We went over some problems with lenses, converging and diverging.

Then spent some time on problems with different indices of refraction. I would usually think of air to water and the behavior if you get mixed up.

Used Snell’s Law and another equation with the velocity of light, c, and the index of refraction.

n= c/v

The index of refraction is about 1 in air and is 1 in a vacuum. It doesn’t (in the context of this class) go below 1.

Preparing for an Algebra Test

We went through the review packet for the test.

One thing to watch for was to only combine like terms, so either numbers or variables with the same exponent.

We saw the difference of perfect cubes more than once, which has a corresponding formula.

You can always multiply by different forms of one to do things like getting a common denominator. Something divided by itself is one.

Need to be careful with parentheses and notation to make the work easy to follow and check.

If the denominators of two equal fractions are equal then the numerators will also be equal, that can save a few steps.

Usually quadratic functions have two solutions, sometimes they have one or none.

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.

Tutoring Trigonometry, Use Trig functions on right triangles!

We went over problems with trig functions.

The functions only work with right triangles.

Looked at both the angle of elevation and the angle of depression in a few problems.

The final two problems were more complicated and involved two equations and two unknowns as well as calculating the tangent function of two angles and factoring as well as multiplying algebraic expressions.

Graphing a hyperbola “neatly”, tutoring Algebra II

We started by graphing a hyperbola “neatly” as the directions stated. That involved finding the vertices, foci and drawing the diagonal asymptotes.

Checked one point by plugging in an x value to see if the graph was accurate.

The sign in front of the variables is important to determine orientation of the conic sections.

The natural number is e and is the base for the natural logarithm. e is approximately 2.7


We mostly looked at

Also finding the positions of angles and the names of equivalent angles.

Less than zero can be read as ‘negative’ and more than zero can be read as ‘positive’.

For these trig functions, the hypotenuse is always positive. And many times, using the unit circle is useful. Although, sometimes multiplying the ratios by a common factor can also be useful.

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.

How do I solve 3^{2x}=81?

A few options. The option on the left assumes that the answer is a positive whole number.

The option on the left would be more useful for a not-whole number.


We started by looking at evaluating trig functions of angles. They involved 30 60 90 triangles and 45 45 90. So we began by looking at those triangles.

It’s easy enough to derive the ratios for the isosceles triangle.

Then we compared that to the other triangle.


Often, it helps to draw the angles on the xy axes starting from the 0° position (usually East). Then connect the angle to the x-axis to make a triangle.

Also, the functions are pronounced ‘sine’, ‘cosine’, ‘tangent’. If you’re going to say them out loud, say those, not “sin”, “cos”.

If you get to “csc”, how would you even say that out loud?