Multiplying Algebraic Fractions uses all the same principles as multiplying non algebraic fractions, it just complicates it a bit more with the addition of variables.

To start, the best thing to always do is check if anything is factorable. It’s important to do this because it makes everything so much easier in the long run.

Once everything is factored, find the non permissible values. These will just be found in denominators, as the denominator cannot equal zero.

Next, it’s pretty simple, all you need to do is multiply across the multiplication sign for both the numerators and denominators. In a way this is even more simple than addition because we don’t need to find a common denominator.

Once you have the multiplied numerator and denominator, just make sure to simplify further if possible.

That’s really all there is to it, the main thing to keep track of is negatives. Negatives can easily mess everything up, so it’s worthwhile paying attention.

Dividing algebraic fractions is essentially the exact same thing, except that before multiplying, we just flip the second fraction. It can never be the first fraction that flips.

This also means that we need to find the non permissible values for both the denominator and the numerator of the second fraction, as we can never have a zero on the bottom of a fraction at any point. -you cannot divide by zero.

Graphing quadratic inequalities is honestly pretty easy to get down as long as you understand quadratic equations and basic graphing, two things we’ve learned before this unit. The trick is just understanding how the two work together and more importantly, why they work together. Here are a few steps:

1. Graph the Quadratic Function:

Start by plotting the quadratic function without even looking at the inequality symbol, that won’t be important until later on. I won’t go over how to graph a quadratic function as I’ve already done that in a previous blog post, but the final result should be a parabola.

2. Find the Shaded Area:

My basic rule of thumb for understanding which inequality will correlate to which region being shaded is simply that less means under and more means over. This works best for me when I visualize it like a linear inequality. I know that anything above the line will be greater than / greater than or equal to that line- the same going for less than / less than or equal to. Now if I imagine that a quadratic is a line simply being bent (which isn’t the full story but no need to overcomplicate right now), it’s super easy to still use that under/over method. If the parabola opens up, the under is the space outside the parabola, and if it reflects down, it’s the space inside, and vise versa for the over.

*It’s important to remember that when it’s “less/greater than or equal to” the line on the graph will be solid, but if it excludes the line, the line will instead be dotted/broken.

3. Test a point 

Finally, it’s good to make a test point on the graph, the simplest being (0, 0). If you can plug it into the inequality, and get a true answer, then you know that’s the space that needs to be shaded. If not, it needs to be empty.

This week we learned all about parabolas which are derived from the base equation: y = x²

When graphing a quadratic equation, there are an infinite number of points on that given parabola that one could attempt to find. There is one point, however, that stands out from the rest: the vertex. The vertex is the turning point of the parabola, and is either the minimum or the maximum depending on if it opens up or down. It sits directly in the center of the parabola which is always symmetrical, meaning the line of symmetry runs right through it.

In the previously mentioned equation y = x², the vertex will simply sit at (0, 0) as it is the unaltered version or the parent function. However, when different aspects of that equation are changed the vertex (among other things) will change position.

For example:

The equation y = x² + 2 will see the vertex move up on the graph by two

The equation y = (x – 4)² will see the vertex move to the right by four

It is easy to see – 4 and think to move the vertex four to the left, but it’s important to remember that it’s flipped. If it were + 4 THEN it would move to the left.

Another important thing to note that does not effect the vertex is the coefficient placed in front of the “x²”

The equation y = 2x² would have a stretch double that of the base parabola. Imagine it like the equivalent of “slope” but for a function that can’t have slope because it is not linear. If the coefficient is negative, it simply means that the parabola is flipped, now opening down.

This all means that an equation that looks like y = -4(x + 2)² – 7 would have a vertex at the point (-2, -7) and would open down with a stretch value of four.

This week, we learned about forming perfect square trinomials. A perfect square trinomial is exactly what it sounds like: a trinomial that is in its self a perfect square that when square rooted gives you a binomial. This mean that a perfect square trinomial can be broken down into (a+b)².

An incredibly simple example of a perfect square trinomial would be:

x² + 8x + 16

It’s easy enough to deduce the answer via simple inspection with this one, but it will help to break down how we know it’s a perfect square.

My first step is always splitting the middle term (8x) into two (4x + 4x). This would then give me:

x² + 4x + 4x + 16

The easiest way forward from here is to simply see if one of those 4’s will square to the final term. In this case, 4² = 16 which means that when we pair this info with the fact that the first term in the equation is a square (x²) yes, we can confirm that this is a perfect square trinomial. Alternatively, I could also just square root the final term (16) and make sure it matches half the middle term (which again, it does).

An easy mistake to make is thinking something like x² + 16x + 16 would be a perfect square, as every induvial term on it’s own is a perfect square. It’s incredibly important to remember that just because on their own they’re perfect square, we’re not talking about the terms, but rather the ENTIRE trinomial.

This week in pc11, we went over exponent rules. To be completely honest, the group I was sitting with completely saved me on this one. Going into the week I’d completely forgotten everything, so being able to work with them helped immensely. That group in particular was helpful because we were actually able to talk to each other, something that isn’t always possible in other groups because no one want’s to talk. One of the helpful things was that all three of us looked at the questions (as well as math in general I think) in different ways, so if one person was struggling, the other two had different explanations that could help. Now, onto the actual topic, fractions as exponents. Before this week, I’d never really understood how these would work, but a one point in conversation with my group, it just clicked in my mind.

How it works in my mind is that each one is essentially two different, distinct steps.

First, take the denominator and use that as the root.

For example, with 8^2/3, we’d take the 3 and use that to find the cube root of 8. That would give us 2, with a remaining 2 in the exponent slot.

For the second step, we’d just treat that remaining exponent like any old regular exponent.

Using the same example, that would just mean we’d square the 2 for a final answer of 4.

While it might seem a bit basic and maybe a bit boring, I think that the number groups we’ve been learning for years now are incredibly important. For the last few years, at the start of every math class, I’ve learned about them, and then like clockwork, promptly forget everything I know about them. Usually, they don’t really ever come into play for the rest of the semester, which means we don’t really have a chance to practice. This past week however, I found that while reviewing/recapping the topic, all the things we learned previously came back to me much easier than before. To explain the concept of the groups, I find it easiest to imagine them as a sort of Russian nesting doll. Within the category of “real” numbers there are two smaller categories, those two being “rational” and “irrational” numbers. Irrational numbers stand on their own and have no smaller group within them, while rational numbers include another group, integers. Within the category of integers, you can find the group of whole numbers, which in turn has an even smaller group, natural numbers.