In Pre-calc 11, I have learned many things. In this blog post, I have chosen my top 5 favorite lessons I have learnt over my grade 11 journey. I decided to write about this because I want to be able to use this as a reflection of my journey.

1. Sine Law

I chose this because the Sine law is applicable to all tringles when you have the right values you can solve for any side or angle.

2. Cosine law

I chose this because  like the Sine law the Cosine law is applicable to any triangle when you have the right values, allowing you to solve for any side or angle.

2. Factoring equations

I chose this because factoring is one of the core foundational skills of Pre-calculus without it you couldn’t do the rest

4. – The high school version of the Pythagorean theorem

I chose this because this new version of the Pythagorean theorem because it caused me to re-imagine trigonometry into axis and coordinates.

5. Rationalizing the denominator

I chose this because this was the reason I was able to transfer quadratics from equations to on a graph allowing me to visualize.

Welcome back to rational expressions this week in Pre-Calc 11 we learnt  how to multiply and subtract rational expressions! I chose this topic because multiplying and dividing rational expressions is the foundation of our current unit. Lets first try a multipication question!

When dealing with rational expressions our first step is to always see if our expression is factorable and if you remember from our previous blog it is recommend to put brackets around the numbers you have checked for factors.

And right now it would be a good time to set our non-permissible values by checking the bottom.

Next we cross out the shared binomials from the top and bottom. Leaving us with 35/3x.

Now let’s try a division rational expression question. As always the first step we take when dealing with rational equations is too factor and, next you say we should cancel out right? Wrong just like a normal fraction division question we have to turn it into a multiplication question.

Now that we flipped it we can cancel out and also a good time to find your non-permissible.

Leaving us with this.

Tada you learnt how to multiply and divide rational equations! Come back next week to finish off rational expressions!

This week in Pre-Calc 11 we learned how to add and subtract rational expressions. I chose this topic because adding and subtracting rational expressions is the foundation of our current unit. Let’s try a question now!

When approaching a rational expression question the first thing you should always do is to see whether the question is factorable or if there is a difference of squares. However in our question here we can do neither so we put brackets around to indicate that we already have checked if it’s factorable.

Next step we just like adding regular fractions, we need to find the LCD o (Lowest Common Denominator) in order to add these fractions this being 3(x-2). Now we multiply the bottom and top then put it into a high school fraction (that one big long fraction).

That now leaves us with this and now we check if we can simplify this expression.

And oh it doesn’t look like a multiple of 10 can get a sum of 5 leaving us with this equation.

Lastly we now need to set a restriction for X. We do this by looking at the denominator in our example here it’s -2 so what cancels out negative 2? Positive 2 right? Meaning x cannot be +2.

Now let’s try a subtraction question.

Just Like an additional question, our first step is always to see whether the expression is factorable or has a difference in squares. In this case we don’t see any here and the denominator has the same variable.

Another rule we have to add before turning it into a high school fraction is that when dealing with binomials, especially subtracting, we should flip the signs. I know that may seem confusing but stay with me now. The reason we do this is because when we turn binomials into high school fractions (the one big fraction we often forget about what the + or – signs apply to what.

Now we put like terms together leaving us with then we set our restriction leaving us being 0 be cause 0 x 0 is 0 cancels itself out.

AND TADA!!! Now you learnt how to do subtract and add rational expressions

This week in pre-calc 11, we learned about sine law and how it can be used to find an angle or a side of a triangle. The law states that in any triangle the ratio of the sides of a triangle and their opposite sine angles are equivalent to each other.

Here is the formula and location of the variables.

Let’s now try an example.

In this example it’s asking us to use the sine law to find side b.

And from our given values we only have Sine A and B. Then we plug the given values into the formula.

Then we solve for the side (b) rearranging algebraically as needed, giving us the answer.

Now let’s try another example.

In this example it is asking us to find angle A.

And from our given values we only have to use Sine Sine A and C. Then we plug in our values again like last time.

Then we solve for the side C rearranging algebraically as needed, giving us the answer.

This week in Pre-Calc 11 we learned how to solve inequalities with quadratics then writing them down into interval notation. I chose this topic because it is something I especially struggled with on the skills check.

The first step in solving a quadratic inequality is to identify our equation. We can tell when a quadratic inequality is a quadratic inequality by x(squared) or 2 x variables multiplying.

Now that we know our inequality is a quadratic, our next step is to find the roots of our equation by converting it into factored form.

Giving us the x-int of the parabola we can now plot it on the line graph.

This week in Pre-Calc 11 we reviewed systems and inequalities and how to solve them algebraically. I chose this topic because it is a skill that I particularly struggled with last year. Let’s try an example.

In this equation you might be panicking a bit because there is no = sign, but fear not the < and > symbols can be treated the same as an equal sign when isolating the variable. So now the first step we should do here is to expand our equation.

Then we move our variables together.

Now we isolate x and simplify and tadaa we did it!

This week in Pre-Calc 11 we learned how to convert equations from general form to standard form using the completing perfect squares method. It is an important skill to understand both equations gives us vital information when solving a quadratic function.

Here are the two different forms

Let’s try an example, the first step is to divide the b variable by 2 then squaring it.

Now add both a positive and a negative of your sum from above, we do this because by adding both a positive and negative it leaves the equation unchanged, canceling out each other.

Now we convert the perfect trinomial into a binomial and tada! We have now successfully converted our general form equation into standard form.

Both forms give us different information, the standard gives us the vertex and the general gives us the y-intercept. 

This week in Pre-Calc 11 we learned about the discriminant. The discriminant is the part underneath the root sign of the quadratic formula (b²-4ac), it helps us determine the nature of roots in the equations that we are working with so here’s how to use it.

So there are 3 solutions you can get from the discriminant positive which means the equation has two solutions, 0 that means the equation has one solution and if the discriminant is negative that means there is no solution.

Here is the first example with the discriminant being positive.

Next here is an example with the discriminant being zero.

Lastly, here is an example of the discriminant being negative.