We looked at slope intercept form and how m is the slope and b is the y- intercept.
To get the y-intercept, you set x = 0
to get the x-intercept, you set y = 0
To get the slope, you use two points (some are easier to work with than others)
Then take (y2 – y1)/(x2 – x1)
Subtracting a negative two is like like adding two
x – – 2 = x +2
If you have the slope and a point, you can plug in the x and y to get the value for b and then complete the equation.
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.
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.
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.
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.
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.
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.