This week in precalc 11 we learned how to add, subtract, multiply and divide rational expressions. In this unit, rational expressions are complex-looking fractions with variables. I will be focusing on how to multiply these rational expressions

Working with rational expressions is no different than working with regular fractions. In regular fractions, we multiply across and simplify where possible.

Example: 

• simplify
• multiply across
• set restrictions (non permissible value)
• cancel things out

Example:

• factor if possible
• cancel things out
• multiply
• set restrictions
• bring down the other expression and do the same

This week in precalc we started working with rational expressions (fractions with variables). We first started with simplifying them which is a key aspect for working with these kinds of questions (adding, subtracting, etc.). It is important to remember after each question you are required to state the non-permissible value because there is a variable in the expression.

Non-permissible value: The value of a variable that makes the denominator of a rational expression equal to zero.

example 1:

• the first thing should always be observing to see if you can make the expression any simple like factoring out anything or cancelling any numbers or variables.
• In this case, we can cancel (a-8).
• The non-permissible value should always be stated before you cancel anything out.

example 2:

• by observing the expression we can see that both the numerator and the denominator are factorable.
• write the non-permissible value
• cancel anything that is left over.

example 3:

• here we can see that the expressions are similar but they aren’t in the same order.
• in this case, we can still simplify BUT it does not simplify to a 1. It simplifies to a -1.

This week in Precalculas 11 we went over inequalities and what they mean. We also learned how to graph them. In graphing inequalities, we can use what ve learned from the last unit, like vertex, x-intercept, y-intercept, etc. to help us graph inequalities.

Example : y> – (x+4)^2

• first I graphed the expression on demos without the inequality sign to see what it would look like. in this example we already have the inequality in vertex form so it is easy to graph but if we dont have to solve it a little bit.

• now that I plugged in the inequality sign there are some changes we can see so let’s break it down

1. As you can see the parabola has a dashed line, this is because our inequality does not have the small line under it. This means that y is greater than… If it was greater than or equal we would but a bold line.
2. Everything outside the graph is shaded. This is because of the inequality sign. Our expression is telling us that our solution has to be greater than – (x+4)^2. So how to we figure out where to shade.

• take a random point in a graph and plug in the x and y into the original function
• if you get a true statement then this means that the shading should go in this zone. In this case it is outside the parabola. 
  

This week in pre-calc 11, we briefly went over inequalities and systems in one lesson. By inequalities, I’m referring to the relationship between two expressions. I like to think about the inequality signs as alligator mouths, they eat whichever number or expression is the biggest.

example: x>7 — x is bigger than 7 that’s why it is getting eaten. X could be any number that is above 7

example: 78 x —– x is bigger than or equal to 78.

• during graphing or on number lines if there is a dash line, it means the inequality sign does not have the small line under it

• if there is a solid line, it means that the small line is on the inequality sign.

Graphing inequalities :

• When the expression is given to us, we move everything to one side and make sure to isolate y. what we are left with is our slope and y-intercept.

• graph the expression
• the inequality sign tells you which side of the line to shade, to figure this out take point (0,0) substitute into the inequality and simplify. If the inequality is false we shade the part that doesn’t contain the point.  if it is true we do shade the point.

General form – Ax + By + C = 0 

Vertex form – Y = a (x-p)^2+ q

Example: y=3x^2+18x+20 

Step 1: Factor out the leading coefficient. Steps 2 & 3: Complete the square, at the same time as completing the square multiply the number that is not part of the equation by the coefficient

step 4: simplify

This week in Precalc 11 we started the intro to quadratic functions. The graph of a quadratic function is a parabola, which is a U-shaped curve. The direction and openness of the parabola depend on the sign of the coefficients. If a > 0, the parabola opens upwards, and if 0>a the parabola opens downwards.

Vocabulary :

parent function:

• the parent function spacing follows the 1 – 3 – 5 – 7 … pattern
• y=x^2 
  

Vertex: the turning point of the parabola (also known as the most important point on a parabola)

Maximum of quadratic function: the highest point on a graph when a parabola opens down

Minimum of quadratic functions: the lowest point on a graph when a parabola opens up

The axis of symmetry: the equation of a line that divides the figure into two equal parts where one is the mirror image of the other.

The standard form: 

• Vertical Translations: Adding a number moves the graph up. Subtracting a number moves the graph down.
• Horizontal Translations: Adding a number inside the function shifts the graph left. Subtracting a number inside the function shifts the graph right.
• Vertical Stretching/Compression: Multiplying the function by a number: If the number is greater than 1, it stretches the graph vertically. If the number is between 0 and 1, it compresses the graph vertically

Here is an example:  f(x)= (x-2)^2+3

The blue parabola is the parent function, and the green parabola is the example. In the green parabola we can see that the minimum value is y=3 but in the blue parabola it is 0. this is because in the equation we have a vertical translation of +3 this made the parabola move up 3 units. Another difference is that the green parabolas axis of symmetry is at x=2 rather than 0. Again this is due to the horizontal translation. Even though it says -2 in the brackets we are moving to the right when a negative sign is seen. If a positive sign is seen in the brackets we move to the left.

This week in Precalc 11, in addition to reviewing our past units, we learned about the Quadratic formula. The quadratic formula allows us to solve any quadratic equation that’s in the form ax^2 + bx + c = 0. This is useful because you can use this formula anytime you don’t want to do grouping, inspection, etc.

In an equation, this is where the ABCs are:

To solve an equation you simply plug in the numbers and solve:

In the end, if you are left with a radical that can be simplified, make sure to simplify. Then you work out the equations till you get your answers.

In this example I have simplified the radical to a mixed radical, the index, the denominator and the constant can all be simplified so we simplify.

This week in Precalculus, we learned how to factor equations with radicals. It’s an important skill since radicals pop up in different units, and knowing how to factor them helps in solving a variety of problems.

Example 1:

step 1: isolate the radical 

step 2: get rid of the radicand by squaring everything. This removes the radical sign and squares the other side. Be careful because if we have a binomial on the other side we need to write it out twice and foil.

step 3: move everything to one side so we have 0 on one side. If your squared variable is positive on one side it’s better to move everything there because negative squared variables don’t look nice.

step 4: factor by prefered method (mine is grouping)

step 5: find out which number makes the equation equal to zero. we might have 1 answer or 2 answers.

This week in Precalc 11 we covered the basics of factoring and we also got to touch on solving quadratic equations using factoring.

What makes something a quadratic equation:

• Quadratic equations are polynomial equations of degree 2.
• General form: ax^2+bx+c=0 
• have to make one side equal to zero when factoring
• there will be up two answers when factoring them
• follows the zero product law: the zero product law states that if the product of two factors is zero, then at least one of the factors must be zero.

Example 1: 

 step 1: Factor – you want to first look for any common variables or numbers, if there aren’t any you can factor by your prefered method (boxing, grouping, inspection, etc) the numbers that are on the other side of the zero. (during factoring quadratic quations we do not touch the zero it just stays there, because essentially the whole point is to get the none zero side to equal zero.)

step 2 : apply the zero product law – applying the zero-product property helps us find the values of $x$ that make each factor equal to zero, thus giving us the solutions to the quadratic equation. For example -8 x 8 = 0

A great way to check to see if you have done your work correctly, is to just plug back in the number to the original equation.

Example 2 : As we can see in this example, we didn’t have all the number on one side and zero on another, but thats okay because all we need to do is to move them to one side. The reason as to why the answers are fractions is because when applying the zero product law, we are taking the equation and isolating x.

Example 3: in this example we are dealing with fractions. After moving the constant to the left hand side, we multiply everything by 6 because 6 is the least common multiple. then we factor, group and apply the zero product law.

This week in Precalc 11 I learned how to factor polynomials by grouping them. This is a good skill to have as polynomials can look scary sometimes and factoring them isn’t easy.

Example: x^{2}+2x+6x+12

Step 1: Start by drawing a square and dividing it into 4 equal boxes inside. Then, place numbers in each box. It’s crucial to arrange these numbers in a specific order so that when you examine them diagonally, the products of the diagonals are equal. Additionally, ensure that similar variables are positioned diagonally opposite each other. For example, if you have 2x in one box, place 6x in the box diagonally opposite to it.

Step 2: Take out the greatest common factor (GCF) or common variable from each row.  This process will result in having two numbers at the top of the box and two numbers at the left of the box. If there’s no apparent GCF or common variable in a row, simply write 1, as 1 is divisible by everything. This step helps simplify the expressions within each row, making further calculations easier.

Step 3: Next, let’s group the numbers together. The two expressions at the top of the box should be grouped together, as well as the two expressions on the left. Ultimately, you should have two binomials formed, where the product of these binomials equals the polynomial we began with.

(x+2)(x+3)