Tutoring Algebra at CSUMB, vertical line test and horizontal line test

We looked at functions and inverses of function.

One quick way to see whether something is a function is to do the vertical line test. If a vertical line can ever pass through two points, it’s not a function.

An inverse function switches the x and y values. You can figure out if a function has an inverse that is a function by doing the horizontal line test.

Looked at telling the shape from a few types of equations, especially lines and parabolas.

Graphing Periodic Functions – Sine, Cosine, Amplitude and Period

We looked at periodic functions.

Started most problems by finding the amplitude and the period.

The amplitude is basically the absolute value of the coefficient multiplying the function. If there is no number, the amplitude of a sine or cosine function is 1.

The amplitude can be found by 2π/# the number in front of the x, as in cos(2x) the # is 2.

After that, you should know the basic shape of the sine and cosine functions. You can graph two periods (for this set of problems) and fill in the zeros.

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.

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.

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.

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?

Algebra, Variables with Exponents Positive, Negative, and to the Zero

An explanation of why numbers to the zero power equal 1.

Dealing with exponents in various situations.