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.

Circles, Ellipses, Hyperbola

We started by looking at graphing ellipses.

Began by with the equation for a circle of radius = 1. Then saw how coefficients change the shape into an ellipse.

The ‘eccentricity’ in a way measures how different an ellipse is from a circle.

Saw how the center is found as well as the axes. X is connected to horizontal and y is connected to vertical.

Ended by looking at hyperbola and a Friday sheet that included Sigma (summation) notation.

How to get horizontal asymptotes and get equations for cubic functions given a graph

We started by looking at problems from the most recent test.

Getting the x and y intercepts just involves setting one variable or the other equal to zero since that is where the equation will intersect the axes.

A quadratic equation will generally have two solutions, it can be less.

To find horizontal asymptotes, I just plug in a large number for x and a negative large magnitude number. You can think of positive/negative infinity or even positive/negative 100.

It helps to know the general shapes of functions, lines, parabolas, cubic functions, etc.

The hardest problem might have been figuring out the equation of a cubic function based on the graph. Two zeros were given as well as another point. That means that one of the terms with a zero would be squared since there were only two zeros.

Why you should move beyond ‘undefined’ and think more in terms of infinity and negative infinity for some problems

Some other problems involved looked at shapes of graphs.

Division by zero should not longer be thought of as simply ‘undefined’. In some of these problems it indicated a vertical asymptote which can be graphed.

The values are closer related to positive and negative infinity.

Besides features like that though, you can simply plot points to get the shape of the equation. But you need to be careful to not cross the asymptotes, etc.

How to find a second complex root when factoring a polynomial using synthetic division

A few of the problems involved synthetic division.

They might start as third degree or fourth degree polynomials with one or more of the roots given. There was an equation that would give a second root if one was a complex number (if all the coefficients of the original polynomial are real).

The complex conjugate will also be a root.

For example, if 1 + i is a root, then 1- i is also a root. (given the right conditions)

If you have two roots and use synthetic division (or long division) you will have two roots and a quadratic equation, which will be much easier to factor.



How parentheses help you keep track of terms and avoid making mistakes

We spent most of the time on a Friday sheet. Including parentheses when the problems get more complicated can be very helpful. In the case of these problems, it was useful to see that multiply parts were being multiplied. Like FOILing.

To find intervals that are valid for inequalities, you can think of the inequality sign as an equals sign to find the boundary points and then test numbers in the different regions.

Another problem involved u-substitution again.

Something new in the homework was the ‘end behavior diagram’ which basically shows whether an equation goes very high or very low as the x-coordinates get high and low.

In which quadrant is the point (-5, 4)?

Saw this question, thought it could be best shown with a visual.

The first coordinate is the x-coordinate, the second is the y-coordinate.


How can DMX unintentionally help you with math? Distinguishing between domain and range

We started with some inequality problems and interval notation. The thing we focused on was square brackets vs curved brackets [ and (. One is inclusive, like a closed dot, and the other is exclusive like an open circle on a number line

When boundary points are found, three regions are divided. It’s good to pick convenient numbers in each region to test them.

We also looked at U notation (union) and the upside down version (intersection). Sometimes that is used. Same meanings as the words ‘or’ and ‘and’.

Domain refers to the horizontal (x-coordinates) a trick to remember that is DMX, if you know who DMX is.

Also plotted a few equations using points.

The sum of perfect cubes equation came into play again.

We looked at how the discriminant can determine whether a quadratic equation can be factored.

And u-substitution was used again.

Greater and ‘Greator’ with ‘or’, Tutoring Algebra II

Today we looked at inequalities, plotting them on number lines and interval notation.

One trick is to use the similarity between the word ‘greater’ and ‘greator’ with an ‘or’, most times inequalities with absolute value and a greater or greater/equal symbol will result in a solution set that includes the word or.

Absolute value is the magnitude, the distance from zero, always positive.

Tutoring Math Friday Sheets at Stevenson School

We worked through the Friday sheet. These sheets are going to be challenging and can include material outside of the normal classwork. I have seen them before for students at RLS.

One important equation to recognize is the sum of perfect cubes and the difference of perfect cubes. One can be turned into the other easily with a little bit of algebraic manipulation.  Probably by looking these up he will memorize them if they continue to pop up, which they likely might. And if you recognize what something is, you can often look it up if you include enough detail “difference of perfect cubes” in Google for example.
With graphing, knowing the basic shape of the graph in advance by looking at the equation can help, especially lines/parabolas at this point.
You often factor when simplifying algebraic equations.