Category Archives: Grade 11

Final Blog Post – Top 5 Things I Learned in Pre-Calculus.

This year in Pre-Calculus was a journey that had its high points and low points. Here are the top five things I learned.

1.) Rationalizing the denominator, It is a concept that is very easy to mess up. Whether it is not multiplying by the conjugate or leaving a negative on the bottom it is something that has to be done with extreme care

  1. take the conjugate of the denominator
  2. multiply both the numerator and the denominator by the conjugate
  3. distribute everything
  4. simplify if necessary
  5. multiply by -1 if the denominator is negative

2.) Geometric sequences with percent values, this is a concept that is very applicable to real-life situations

If you get a 5% raise you use (1.05)

If something depreciates by 20% you use (0.80 or 1-0.20)

3.) Graphing quadratic and linear equations, one of the most important skills in Pre-Calculus 11, because it appears in multiple units, it’s involved in absolute values, reciprocal functions, and inequalities.

4.) Completing the square with fractions, one of the concepts I struggled with the most.

(I had to work out my own strategy for this)

  1. put brackets around everything with an “x”
  2. to find the square number take your fraction, divide by two then square that number
  3. bring it out and multiply by the coefficient
  4. take your circle number and put it on the square

5.) Graphing reciprocal functions

1. plot your original parent function

2. locate asymptotes that will be at y=1 and y=-1

3. graph the hyperbolas

 

Week 17 – Trigonometry

This week we learned some new concepts for trigonometry.

Last year we used SOH CAH TOA

Now we expanded this concept so we could find side lengths and angle of other triangles than right angle triangles.

We also learned about…

  • special triangles
  • sine law
  • cosine law
  • reference angles

New formulas we learned…

The biggest concept we learned is sine law and cosine law. There are two different situations where they’re useful.

Sine law is useful when you got a side and its corresponding angle paired with it.  It doesn’t what pair (If it’s not the desired angle or length it’s okay).

Cosine law is useful when we have a side length or an angle but when don’t have both.

 

Week 15 – Solving Rational Expressions

This week in math we learned how to actually solve the equation (solving for x)

 Here’s how to do it.

for example number one, 

  1. I cross multiplied to simplify
  2. I got my variable on one side of the equation
  3. I isolated “x”
  4. REMEMBER TO LIST NON-PERMISSIBLE VALUES

for example number two, it’s a bit more complicated

  1. Get the factor to have a common denominator
  2. Instead of making just the denominator common multiply everything by the common denominator (Numerator and Denominator)
  3.  Cancel out zero pairs (when a factor in the numerator is equivalent to the factor on the denominator)
  4. Expand (FOIL)
  5. Get variables on one side of the equation.
  6. Isolate for “x”
  7. LIST NON-PERMISSIBLE VALUES.

Things to remember;

  • Only cross-multiply when there are two values.
  •  Make sure to cancel out zero pairs
  • Multiply the whole thing by the common denominator

 

Week 14 – Multiplying and Dividing Rational Expressions

In earlier previous grades we learned to solve and simplify rational and irrational expressions and equations. A lot of people panic when they see expressions that include fractions, myself included. But it doesn’t need to be hard.

First, vocabulary learned in previous years.

  • Denominator
  • Numerator
  • Multiplying the reciprocal
  • Simplifying
  • Equivalent expressions
  • Factoring

All this vocabulary and concepts are still useful. But now we are including fractions!

By now we have learned how to factor expressions, if not remember

Common

Difference of squares

Pattern

Easy

Ugly

Example;

There is a new concept that relates to previous a pre-calculus unit similar to restrictions, known as non-permissible values. Non-permissible are basically x values that make the expression untrue or make it equal to nothing. If you did 6/0, you wouldn’t get an answer

Week 12 – Solving Absolute Value Equations Algebraically

A couple of weeks ago we learned how to solve absolute value equations by using algebra.

 

Verifying is how you can check for extraneous roots (Roots that give you a different answer). Make sure you remember it’s in absolute value form, so there’s never any negative values, if there’s ever any negative  you know that there are no solutions. If the left side does not equal the right side it’s an extraneous root.

Week 13 – Graphing Reciprocal Quadratic Functions.

Last week we learned about how to graph reciprocal functions.

The steps to solve.

1.) Graph the parabola

2.) Locate the invariant points

3.) Find the asymptotes

 

Three types of Quadratics

Positive Slope, Positive y-intercept

 

It never touches (0,0)

so no invariant points

 

Positive slope (can be negative), y-intercept=0

The asymptotes are touching the parabola

Positive or Negative Slope, opposite y-intercept

 

 

 

 

Week 11 – Graphing Linear Inequalities.

This last week we learned how to graph different types of equations

Grade 10 question

y=x+7

Now we are looking to solve this as an inequality

y>x-4

Right now we have 50% chance of getting it right

But by testing the inequality by putting 2 points in as a check (verify)

I like to use the points (0,0)

y=x-4

0, 0-4

0>-4 (CORRECT)

so the side would shaded in including (0,0)

 

Week 9 – Modelling Problems with the Quadratic Equation.

Find two integers with a sum of 36 and the greatest possible product.

Find the variables

I personally like to use the variable “x”, and since there are two integers that means there are two integers. That means the second variable is (36 – x). For example, if the first variable was 4 and using 36-x, would result in 36 – 4, that means the second variable would be 32.

Variables  

1st x= x

2nd x= 36-x

when it says the greatest possible product, that means that we’re looking for the maximum value.

We put the variables in the factored form.

(x) (36 – x)

1st x = 0

2nd x = 36

Now we need to find the vertex

(1st x) + (2nd x) / 2

0 + 36 /2= 18

(18,_)

Then we input the new “x” value

(x) (36-x)

(18) (36-18)

(18) (18)

=324

So the answer is (18,324).

 

Week 8 – Properties of a Quadratic Equation

This week we looked at graphing our quadratic equations. Even looking at a simple equation can help to see the properties of the parabola. A parabola is a shape that forms.

 

y= x^2 + 6, means that the vertex moves up on the y-intercept, another thing we can analyze the equation is the coefficient, if the coefficient was negative then the parabola would be facing down, also you can see that the coefficient is equal to one, that means the parabola will stretch, if the parabola was less than one, the parabola would be compressed.