This week in PreCalc 11 we learned how to multiply and divide rational expressions.

Rational Expressions are fractions whose numerator and denominator are both polynomials and this week we learned how to multiply two rational expressions together. Typically when we multiply two regular fractions together, we multiply the top numbers which turns equals the numerator and multiply the bottom numbers which equals the denominator. Then at the end we simplify the remaining fraction. The same idea goes for multiplying two or more rational expressions together.

Example:

Explanation:

Step 1: completely factor 

Factor all the polynomials present in the rational expressions. All factors should be completely factored.

Step 2: name non permissible values

Typically with regular factors, the denominator can never equal zero. The same concept goes for rational expressions, the denominator can’t equal zero. That is why if there is ever a variable on the denominator it can’t equal any numbers that would make the denominator zero. In this case, if x equals 1, -1, 9, -9, 0 then the denominator would equal zero so that’s why we must put that x cannot equal these numbers.

Step 3: simplify

To simplify, cross out any factors that are the same on the top and the bottom.

Final Answer:

The final answer will equal all the numbers that remain on top multiplied together to equal the numerator, and the same goes for the denominator.

This week in PreCalc 11 we learned how to graph absolute value functions.

Absolute Value Functions is a function that contains an algebraic expression within absolute value symbols. When we learned about Absolute Values before, we learned that whatever number is between the absolute value symbols must come out a positive. For example, | -3 | = 3 . Absolute Value Functions are similar in the sense that the y-values for Absolute Value Functions must be equal to or greater than zero.

Example:

y = | -5x + 10 |

Step 1: Graph the Parent Function

The first step to graphing this Absolute Value Function is to graph the parent function. The parent function is the same function but without the absolute value symbols. In this case, the parent function is y = -5x + 10.

Step 2: Move the Negative Y-Values

As I mentioned before, y-values in the parent function can’t be less than zero which is where the absolute value function comes in. As I mentioned earlier, | -3 | equals 3 because they’re both the same distance from zero on the number line. This relates back to absolute value functions because the negative y-values for the parent function, when put in the absolute value symbols, is a positive. So all we have to do is change the negative y-values into a positive and then graph it again.

That is how you graph Absolute Value Functions

This week in PreCalc 11 we learned how to solve systems of equations algebraically. A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. The systems can either be both linear, be both quadratic, or be one linear and one quadratic. The reason for finding the unknowns (typically x and y) is because it’s the point that, when both the equations are graphed, they intersect. If the and then the point the equations intersect when graphed is But sometimes we can’t graph both the equations to find the solution(s). So to figure it out, we have to solve in algebraically by using a method called substitution.

Example:

Step 1: Isolate One Variable

The first step to solving a system is to make one of the two equation equal either one of the two variables. In this example, one of the equations is equal to y. To make things easier you can make the simpler equation equal one of the variables, too.

Step 2: Substitute 

Now that we have one equation equal y we can place that equation into the other equation. At any place of the equation that I see y I’m going to place the expression that y equals

Step 3: Solve for X

or

Step 4: Substitute

Now we have two x- coordinates that could lead to the solution. There is possible to have two solutions, but even if we have 2 x-coordinates we could only end up with 1 solution. With the 2 x-coordinates that we have, we put them back into the other equation and solve for y

Step 5: Verify

To make sure and are real solutions, we should plug both values into both equations.

Since both solutions make both equations true then that means they are real solutions and that is where the two equations intersect on a graph.