This week we learned a lot about the substitution method.

The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Substitution method can be applied in four steps

Step 1:

Solve one of the equations for either x = or y = .

Step 2:

Substitute the solution from step 1 into the other equation.

Step 3:

Solve this new equation.

Step 4:

Solve for the second variable.

Example 1: Solve the following system by substitution

Solution:

Step 1: Solve one of the equations for either x = or y = . We will solve second equation for y.

Step 2: Substitute the solution from step 1 into the second equation.

Step 3: Solve this new equation.

Step 4: Solve for the second variable

The solution is: (x, y) = (10, -5)

This week in Precalc 11 we learned how to solve a quadratic inequality with one variable using a sign Chart.

The steps when using a sign chart is to:

1. Factor expression

2. Determine zeros

3. Use a sign chart for each factor

take this equation and make it equal to zero the  factor —–> 5x > 2(x2-6)

Once you have factored determine the zeros (-3/2 and 4) and place them on two number line. Each number line will represent a different factor. One will be (2x+3) and one will be (x-4).

For each factor do a test point in all 3 sections. Pick a point less than  -3/4, between -3/4 and 4, and greater than 4.  Plug these test points in to the factor and determine whether the outcome is negative or positive.

Add up the positives and negative of the two number lines to determine what the equation will finally equal.Since the equation is asking for the equation to be less than zero we want x to be negative.

3 things I learned this semester thus far:

The number in front the square root sign is the coefficient, the small number connected to the square root is the index and the number inside is called the radicand. We learned that the index is a very important clue for the equations because it tells us is the number in the radicand spot can me both negative and positive or only positive. Even number index can only be positive while uneven index can be either. Also simplifying equations, the index is important because it tells us what we can take out to become the coefficient. if the index is 2 we take groups of two.

Common Factors

Difference of Squares

Pattern

Easy

Ugly

C- Common Factor is when we look to find if the trinominal has something in common and if it does we will take it out so it makes our trinominal look more simple and easier to factor. For example in the picture below we need to recognize that we can take out a common factor of 3.

D- Différence In squares we need to see 3 things. Is it a binomial, is the refond term negative and, are they both perfect squares. If all of these are yes then we can factor it quite easily like in the example below.

P- Pattern is probably the easiest one because we are just checking if it has the right structure, and if it is then it is most likely facorable. Does it have X squared, does it have an X term and a number at the end. If yes then you can proceed to the next step.

E- Easy means if it is a easy polynomial to factor as in is does the X squared have a coefficient of 1. If it does this will be easy. Don’t stress until they are ugly.

U- Ugly is the opposite of easy (of course) which means that is the X squared has a coeficcisnt of 2 or more then it is considered ugly and you’ll have to do a bit more thinking.

completing the square-

probably the most complicated way to factor in my option because in order to do this process you need to move the third term and bracket the first 2. Then half the square the second term, the one inside the bracket will be positive and then one outside the bracket will be negative. Keep in mind if the first term has a coeficial you have to times the number by the coeficiant outside the bracket. To made this more clear I will show you a photo below.

This week in precalc we learned many things. We had to remember many things in order to successfully graph a quadratic equation.

• Convert between standard, general and factored form
• Standard form  (vertex form) – vertex, line of symmetry, stretch value, opens up or down
• General form – y intercept (0,y), stretch value, opens up or down
• Factored form – roots (x- intercepts) (x,0), stretch value, opens up or down – > line of symmetry (find the average of the roots)  ->the x value of the vertex
• ** factored form there needs to be integers rather than fractions
• a, p and q transform the graph
• a->stretch value, negative -> opens down, positive -> opens up
• A is a proper fraction 1/2 – compression – gets fatter
• A Is larger than 1 – stretch – gets skinnier
• P – horizontal translation (slide) (x-2) -> right  (x+5) …. (x –5) -> left
• Q – vertical translation (slide) positive – > up   negative -> down
• Maximum (opening down) and minimum (opening up) …. Vertex (y value or q)
• Standard graphing pattern … 1 3 5 ….  So if a = 2     2  6 10
• Domain …all the values for x that you could plug into the formula …….

X E Real

• Range – all the possible y values    y > minimum or  y< maximum
• Completing the square  – changes general form into vertex form
• Factor a trinomial …if it factors – nice roots
• If doesn’t factor …. Quadratic formula … discriminant …
• Table of values to graph
• Develop an equation from a word problem. modeling

These are key things in order to succeed in this chapter.