We learned a lot of things this year in pre-calc 11, but there are a few standout things i think are especially important and/or interesting.

  1. The Discriminant
  2. Completing the Square
  3. Sine / Cosine Law
  4. Absolute Values
  5. Graphing Parabolas

The Discriminant

The discriminant is a part of the quadratic formula that tells you how many roots/solutions there are.

It is b^2 - 4ac

A, B and C all come from the quadratic equation you are trying to solve.

So for example if your quadratic is x^2 + 3x - 14  , you would then use the coefficient of the first term as A, coefficient of the second term as B, and the third term as C.

So:

A = 1

B = 3

C = -14

If the discriminant is positive there are two solutions.

If the discriminant is zero there is one solution.

If the solution is negative there is no solutions.

Completing the Square

Completing the square is a method of solving quadratic equations.

When factoring is not possible, you would then look to complete the square.

When completing the square, you take half of the coefficient of the second term and square it.

You then add it in as a zero pair to create a perfect square trinomial.

You can then factor that and collect like terms to reach what is known as vertex form, which can allow you to find your roots.

Example:

x^2 + 4x + 11 = 0

 

x^2 + 4x + 4 -4 + 11 = 0

 

(x+2)^2 + 7 = 0

Sine / Cosine Law

When you have a non-right triangle, but you are trying to find an angle or a side length, you need to use sine or cosine law.

The sine law formula is \frac {a}{sinA} = \frac {b}{sinB} = \frac {c}{sinC}

a, b, and c are the side lengths and A, B, and C are the angles.

As long as there is one fraction with both variables filled and another with at least one, you can use those to then solve to find the value you are looking for.

When there isn’t one fraction with both variables filled, you must use cosine law.

The cosine law formula is a^2 = b^2 + c^2 - 2bc cosA

This also allows you to find any side lengths or angles you need on a non-right triangle.

Absolute Values

Absolute values are special. No matter what you do, everything inside of the absolute value |x| symbol will always, always be positive.

Here are some examples:

| -2x - 11(4) |

 

| -46x |

 

46x

What these symbols do is make the number the principal square root, or in other terms the square of the square root.

This always results in a positive answer, unless other math done outside of the absolute value symbols changes that.

Graphing Parabolas

When graphing a parabola there are a key few things to remember.

First you will generally want your expression in vertex form.

If you do this then you know the vertex, and the scale factor which is plenty to graph it.

Here is vertex form: y= a(x-p)^2 + q

So here is an expression in vertex form: y = (x-4)^2 + 5

Time to dissect the information we got from this expression.

The a value is the scale factor, which is one, so it will follow the scale factor of the parent function, 1-3-5.

Also, the p and q values are the vertex, except the p value has its sign changed.

So the vertex is (4,5).

Now we can just plot our vertex and go up by our scale factor and we will have a parabola!