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.

Knowing a little can help a lot (to look things up) in math

Tutoring precalculus,

We started by looking at problems with sets of equations.

The first were sets of linear equations.

Others had quadratics. For one problem, if you recognized part of the equation was a perfect square, it made the problem easier.

Other times a single mistake like reversing a negative sign could cause problems later on, especially if done early on that means redoing a lot of work.

After the winter break he was a little rusty with a few things.

It’s important to be able to recognize the equations of parabolas, ellipses, and hyperbolas. If you know what something is, you can look up equations for it more easily.

“What do nanometers measure?”

You can measure anything with length in nanometers.

The real question maybe should be ‘what do nanometers measure conveniently?’

For example, you could say your height in meters. You could also say it in nanometers.

But to say it in meters, probably gives you a number somewhere under 10. Which is a convenient number to think about.

You can measure something much longer in meters, say the distance from California to NY. But at that point, you’re talking about a very large number and it becomes more convenient to use kilometers since then you can use a smaller number of that unit.

With nanometers, it’s convenient for things like the wavelength of visible light and things at the atomic level.

“How many Planck lengths are in an inch?”

You can ask Google questions like this if you know how to format them correctly.

“How do I find the angle of a right triangle knowing the ratios of the sides?”

You can use an inverse trig function of the ratio using SOHCAHTOA. Then you can subtract the two known angles from 180° to get the remaining angle.

Why is “the” square root always positive

Since there are often multiple answers to quadratic equations. That is the important thing to remember in mathematics, I think.

‘The square root” implies something singular and refers to the positive, principal square root. That is more semantic and personally, I think less important practically.

There are situations in which negative numbers do not make sense so it does make sense to only think of a positive value for the square root. But not all situations.

Personally, I would rather see two different symbols for square roots in general and for only finding the principal square root. I would like to see a new symbol for the square root finding the principal and have the usual symbol imply multiple solutions.

But I don’t get to decide these things.

Maybe the idea of a different symbol could catch on though.

“What is the compliment of a 32 degree angle?”

Here’s a diagram to show what a complement means for angles.

If you have two angles that add up to 90° then they are said to be “complementary”.

One of the angles in this example is 32°.

“How do I find the sin of any angle?”

How do I find the sin of any angle?

For example, if I wanted to find the sin of 30 or 10, how would I go about doing that.

You need to know whether the angle is specified in degrees or radians since there are two systems for measuring degrees.

At that point, if you can use a calculator, you might press 30 then the sin button.

Or for a different sort of calculator you would input sin(30) then press enter.

Make sure that you are in degree mode or radian mode depending on what you’re working with.

Without a calculator, some angles are easier to find than others. For example, 30°, 45°, etc are triangles with well known ratios of their sides. So you can think of SOHCAHTOA.

For angles related to those, you can use things like half angle formulas.

From the two examples you gave, sin(30°) = 0.5. sin(10°), I’m not sure offhand.

There are tables you can look up values. If you end up looking up a certain value a lot, you may remember it.

“Why are there 9 square feet in 1 square​ yard?”

Since a yard is equal to three feet. Here is a way to see it visually.

Always take the chain rule for derivatives

Sometimes, if you take the derivative with respect to the variable in the expression (and there is only one variable), you can ‘not’ use it since, for example, dx/dx = 1. But in general, you use the chain rule for derivatives.

Reviewing for the final, we started with the earlier version of a derivative which involved limits.

The letter ‘a’ is used sometimes to denote a constant.

With multiple variables, the chain rules becomes even more important.

The general approach for a problem involving changing dimensions of a rectangle was to find an expression for a particular characteristic and then take the derivative. Certain rates were known in the problem.