Last week in Pre-calculus was the 11th week. Sorry for the one-week delay.

One thing that I’ve learned is solving linear inequalities in one variable.

Overall, this topic should be quite straightforward. You just need to be careful of your signs (+/-) and the inequality that you need to use.

First of all, since we’re only solving in one variable, we don’t need to use the x-y graph, just a number line would be fine.

These are all the inequality signs that we’re going to use:

Keep in mind that if you graph them on the number line, there’s a slight difference between greater/less than and greater/less than or equal…

If you graph an inequality with greater than or less than signs, the number that it’s based on is NOT shaded or part of the solutions.

On the other hand, if you graph it with greater than or equal or less than or equal signs, the number it’s based on IS shaded or part of the solutions…

(greater/less than – left) (greater/less than or equal – right)

NOTE: keep in mind that if you DIVIDE or MULTIPLY by a NEGATIVE, the inequality sign must be SWITCHED! check example 2!!

Examples:

We dealt with a lot of things this week, especially about systems. And for this blog, I’m going to talk about linear-quadratic system.

Let’s have a quick recap:

• Linear equation deals with a straight line on the graph. Basically, an equation of a line.
  • The equation being known as y = mx + b, where m is the slope, and b is the y-intercept.
• Quadratic equation deals with a parabola on a graph, with the equation having at least one squared variable.
  • The standard equation being known as y = a (x – p)² + q. Find out more on my blog post about quadratic equations!
• Together, they form a relation called System of Linear and Quadratic Equation. 

Systems (where they intersect) of these two equations can be find out or solved:

• graphically.
• algebraically.
  • substitution.
  • elimination.

However, in this blog I’m only going to talk about solving it graphically.

(At this point, you should be able to graph both of them!)

NOTE: There are three possible cases that can happen if you finished graphing them!

(graphs used are in courtesy of desmos!)

• Two solutions – if the line passes on two points of the parabola
• One solution – if the line passes only on one point of the parabola.
• No solution – if the line and the parabola doesn’t cross at all.

As our math midterm is getting near, this week has been a review week which has been a great refresher for my memory. It was good and bad to know that I’ve forgotten a lot of things. But it’s a good thing to reflect to when studying!

Not much was learned, but rather a lot of refreshers. Here are some of them:

I wouldn’t go into much details about them all as I’ve posted them already in my edublog, but I would do some examples. Here are the links to the blog posts:

• absolute values
• arithmetic sequence
• adding/subtracting radicals

Absolute Values:

• how far away a number from zero is in a number line.
• 
• The answer will always be positive inside the absolute value sign ” | | “
• Can act like a parenthesis.

Examples:

| 5 – 7 |

= | -2 |

= 2

-|-4|

= – | 4

= -4

Arithmetic Sequence:

• A sequence of numbers that are added or subtracted by the same value.
• Addition of a number sequence. E.g. 1 + 2 + 3 + 4 + 5 + 6….
• 
• 
• n = the number/amount of terms you’re calculating
• a = the first term
• tn = last term
• d = common difference

Example:

2 + 5 + 8 + 11…

n = 20

a = 2

Since we don’t have tn, we’ll be using the second formula.

Adding and Subtracting Radicals

• add/subtract radicals with the same index and radicand.
• Simplify if possible.
• 
• n is index, and x is radicand.
• Never add radicand and index. Just add the number outside of the radical.

Examples:

can be added because they have the same index and radicand…

is simplified to as…

can’t be subtracted because although they have the same radicand, they don’t have the same index..

Something I’ve learned this week is about modeling. No – not the modeling one where you pose for a picture – what I’m talking about is creating a quadratic equation based on a word problem, like regarding revenue, money, finding numbers, projectile motions, etc.

There’s actually no need for me to explain what it is as it’s pretty straightforward. All you have to do is analyze the word problem carefully and try and make an equation (of course, with a graph!) based on the problem.

Now, let’s take a look at this problem.

Find two integers that has a sum of 16, and the greatest possible product.

two numbers that has a sum of 16. In other words, it can be written like this algebraically,

x + y = 16

which then, we can arrange into…

y = 16 – x

Now, let’s say that

and…

$latex 16 – x = 2^{nd} number$

Now, if we add those two, we should get a sum of 16.

Why? Well, since our first number is x, and we arranged the equation to become y=16-x, which then means that y has the same value as 16-x, then if you add

x and 16-x, you should get 16 as in the first equation you’re adding x and y.

Anyways, since we also need to find the greatest possible product, we’re gonna multiply them instead of adding, which we’ll get the equation…

Right now, it’s in the factored form, so it means we already have our x-intercepts or roots, which are:

x = 0 ; and x = 16

Now what? Well, we need to find the value of the vertex, because the x-value is the value of each two integers that we need to find and the y-value is the product that we also need to find.

we can find the x-value of the vertex in two ways: graphing and calculating.

For calculating, we just need to find the average, which is pretty easy.

0 + 16 = 16

16 / 2 = 8

And x = 0 is your vertex’ x-value.

Now, for the graphing, we just need to sketch it. but since we already have the x-value let’s do a sketch…

Now, how do we figure out the product? Well, just plug the x-value of vertex in your formula.

f(8) = 8 (16-8)

f(8) = 8 (8)

f(8) = 64

Well, there you have it!

Two integers that has a sum of 16 is 8, and 8 with the greatest possible product of 64!

***All graphs used are taken from desmos.***

Another week in Pre-Calc 11, another blog post!

Something that I’ve learned this week is about the equation

So, what exactly is this equation and what does this do?

Now, let’s do some basics. For example,

If you graph it, it will be like this:

Let’s analyze some parts…

The vertex, which is the lowest or the highest point of the graph, depending on where it is. Where, as you can see in the graph, is in (0,0)

We can also find the domain (x values) and the range (y values) with our vertex.

Always remember that our x-values are always an element of real numbers, or x ∈ R. Why? Because take note that our graph widens horizontally, and take note that if it goes on forever, then our x values can be anything.

Not the same thing goes for our range, though. As you can see, our graph has either a maximum or a minimum, depending on the equation (which I’ll go through later at Section III). And in this case, our minimum is y=0. So it means that y must be greater than or equal to zero, or y ≥ 0.

There are many points to consider such as x-intercepts, which we know as where the line crosses the x-axis or if the equation is y=0. And the y-intercept, which we know as where the line crosses the y-axis or if the equation is x=0.

And we can see that in the graph, both y and x-intercepts are (0,0).

We also need to take note of the line of symmetry.

It’s basically the line that divides the graph equally, which in this case is also x = 0.

Moving on to the main topic, what exactly is

In this blog post, we will divide them first into three sections to understand them more.

the three sections are:

I.     y = x² + q

II.    y = (x – p)²

III.   y = ax²

Section I: y = x² + q

Let’s go briefly over this.

So what ‘q’ does in our equation is translating (or sliding, shifting,…) our graph vertically.

Just note that -q slides downwards and +q slides upwards.

i.e. comparing q = 3 and q = -3

y = x² + 3

y = x² – 3

As you can see, if our equation is y=x²+3, our graph slides upwards by 3, and if our equation is y=x²-3, our graph slides downwards by 3.

Section II: y = (x – p)²

What ‘p’ does to our graph is slightly the same as ‘q’, however it translates or shifts/slides the graph horizontally. Another thing is that this equation is very confusing, because as we are familiar to, we move to the right if it’s positive, and we move to the left if it’s negative. However, this is the exact opposite. If it’s (x-p)², then it slides to the right or positive side of the graph. And if it’s (x+p)², then it slides to the left or negative side of the graph.

i.e. y = (x – 2)² vs. y = (x + 2)²

y = (x – 2)²

y = (x + 2)²

As you can see, if our equation is y=(x-2)², our graph would translate to the right, and if our graph is y=(x+2)², our graph would translate to the left.

Section III: y = ax²

Now, this part of the equation is a lot more different the other two.

What a does in the equation is changing the wideness of the graph, and where the graph opens up, or the reflection depending on its sign.

To make sense out of it better, let’s analyze these graphs.

y = 1/2 x² (TOP)

y = -1/2 x² (BOTTOM)

y = 2x² (TOP)

y = -2x² (BOTTOM)

What we can conclude with that is if the sign (-/+) of a is positive, then it opens up, or the graph has a minimum value or “lowest point.” And if the sign is negative, then it’s opening downwards, or it reflects, or you can say that the graph has a maximum value or “highest point.”

Also, what we can conclude from those graphs is the wideness of graph. We can see that if it’s greater than zero but less than one (0 < a < 1), then our graph will be wider. And if it’s greater than one (a > 1), then the graph will be narrower.

Now, let’s try applying it to the equation y = a (x – p)² + q

So basically, the summary of what we did is:

a = changes the wideness and the direction the graph opens up. (negative reflects down, positive opens up) (0 < a < 1, wider graph) (a > 1, narrower graph)

p = translates the graph horizontally. (negative translates to the right, positive translates to the left)

q = translates the graph vertically. (positive translates upwards, negative translates downwards)

Example: y = 2 (x – 3)² – 4

Applying our skills, we know that a is 2, which means our graph will be narrow and opens up.

We know that our p value is – 3, which means it will translate to the right by 3.

We know that our q value is 4, which means it will translate downwards by 4.

That’s everything! Thank you.

Late post, but in this blog, I will talk about what I’ve learned last week (week 7) in Precalc 11!

To briefly go over it, one of the things I’ve learned is about the discriminant. The discriminant is a part of the quadratic formula used to find how many “roots,” “solutions,” “zeros,” or what we usually call as “x’s” in our formula.

Before we go over that, let’s talk about the quadratic formula quickly.

 this is the formula used to find the “x’s” of a quadratic equation. This is usually used when you can’t “factor” your equation.

Anyways, the “discriminant” that I was talking about is the root part of the equation or “b² – 4ac”

What does it do?

Like I’ve said, it’s used to find if there are any roots or solutions in your equation and how many there is(are).

So how exactly do we determine how many “roots” there is?

*NOTE: Examples used are easier to factor just for easier explanation, but typically you use the quadratic formula if there is no way of factoring it.*

Situation one: b² – 4ac > 0

When the discriminant is more than zero or positive, then equation ax² + bx + c = 0 (quadratic) has two real, unequal roots.

i.e. x² + 5x + 4 = 0

a = 1; b = 5; c = 4

b² – 4ac > 0

5² – 4(1)(4) > 0

25 – 16 > 0

9 > 0

So this means that x² + 5x + 4 = 0 has two unequal, and real roots.

Notice that if you’d factor it, it will be (x + 4) (x + 1), in which your roots are

x = -4 and x = -1, that’s why you’d have two real, unequal roots.

Situation two: b² – 4ac = 0

When the discriminant is zero, then there will be two real and equal roots. Technically, there is only “one” solution because both x’s have the same value. This usually occurs in perfect square trinomials.

i.e. x² + 4x + 4 = 0

a = 1; b = 4; c = 4

b² – 4ac = 0

4² – 4(1)(4) = 0

16 – 16 = 0

0 = 0

This means that the equation x² + 4x + 4 = 0 only has one real root. But notice how you’d know it immediately if you factor a perfect square trinomial.

x² + 4x + 4 = 0

(x + 2) (x + 2) = 0

x = -2; x= – 2. They just have the same value for both solutions, so technically, even though there are two solutions, we can say that it only has one.

Situation three: b² – 4ac < 0

When the discriminant is less than zero or negative, then the equation has no roots or solutions.

Why? Well, think about it. The quadratic formula is like this: r

And we use the discriminant part, which is under a square root sign to figure out the roots.

So, if your discriminant is negative, then it will be an error because you can’t calculate the square root of a negative number.