This week in Pre Calc 11 we learned how to use the Sine and Cosine laws and when to use them. We learned that the Sine Law is used to determine a side or an angle of a triangle in which you couldn’t use SOH CAH TOA, a non-right triangle. Sine Law has two versions of the formula, one is used to find a missing angle and the reciprocal of the formula is used to find a missing side. The formula to find an angle is: The formula to find a missing side is:  To know which of the two formuals to use, we must remember that the variable must be in the numerator.

For example: Determine the measure of angle C.

First we must use the formula with the angles in the numerator to be able to determine angle C: Next we must eliminate the portion of the formula that does not give us any useful information:  Once we have our equation, we can finish solving for angle C:  This can be simplified to: Next we must do the inverse sine of 80 degrees to solve for angle C. Our final answer will be:

The Cosine Law is used when you need to find a third side of a triangle, when the angle opposite to the side is given you are able to use the Cosine formula: to determine the length of the third side. We can rearrange the formula to solve for any variable, however the variable that is written on the left side of the equal side and the cosine variable must remain the same variable. Solving for a side using the Cosine Law formula is fairly straight forward, the first step is to input the information that we were given, into the formula. From there, we must solve the equation and finally square root both sides to find the final answer. We also learned that there is a second version of the Cosine Law formula that is used to determine a missing angle in a triangle:

This week in Pre Calc 11 we learned how Solve Rational Equations. We learned that there are multiple ways to solve rational equations, that vary depending on what the arrangement and difficulty of the equation. The first step is to always verify that our terms are completely factored and if they are not, you must factor it and then determine the non permissible values. The first way to solve a rational equation is to move like terms to one side of the equation for example in the equation: you would first want to move the to the right side of the equation to be able to solve it in a more efficient way, because they have a common denominator you are allowed to subract 3 by 2. Another way to solve rational equations is by cross multiplying. However, this method only works if there are only two fractions, one on either side of the equal sign. For exmaple, is an example of an equation that you are able to cross multiply to help you solve it. To cross multiply, you multiply and leaving you with as your new equation. We also learned how to multiply through an equation using a common denominator. When you do this, you are putting the entire equation over a common denominator which means that it can basically cancel out when you multiply through, which will leave you with only the numerator to solve. For example, can be easily solved if you multiply each fraction by the common denominator which is This will leave you with Once we have cancelled out the numerator, you can now solve for The last method that we learned was that if the numerator’s or the denominator’s of an equation that has only two fractions (one on either side of the equal sign) are equal to one another than that means that the numerator’s or denominator’s must also be equal. This allows you to eliminate the numerator’s or the denominator’s to make it easier to solve. For example, have the same denomintor which tells you that the numerator’s will also be equal to one another, allowing you to eliminate the denominator’s, leaving you with a simplified equation of From there you must determine whether or not it’s quadraitc or linear and then you can solve for If it’s quadratic you must make the equation equal to zero and then you must factor the trinomial and find out the possible solutions for and if it’s linear you must isolate for to determine the solution. Once you have determined the solution(s), you must check to make sure that they are not any of the pre-determined non permissible values.

This week in Pre Calc 11 we learned how to add, subtract, divide and multiply rational expressions. We also learned how to determine non-permissible values of rational expressions. In class this week, we reviewed that a rational expression is an algebraic expression that can be written as the quotient of two polynomials. Rational expressions can not have a denominator of zero, therefore the variable in the denominator can not make the denominator equal to zero. The values for the variable that make the variable equal to zero are called the non-permissible values. We may have to factor out the denominator, to be able to determine the non-permissible values.

To multiply rational expressions, we can either multiply straight across, meaning that we multiply the numerators together and the denominators together and then simplify or we may simplify first so that the numbers and variables that we are working with are smaller. For example, in the expression I would first simplify the expression, by eliminating the from the numerator and denominator of this expression and I would also simplify the and to become and Leaving me with the newly simplified expression, Next we are able  to multiply straight across giving us our final answer: This is our final answer because it can not be simplified any further.

To divide rational expressions, we must first flip the numerator and the denominator of the second fraction, then we are able to simplify and then multiply straight across. When writing the non-permissible values of a rational expression that requires division, we must indicate all of the values that the variables can not be, meaning the ones that were in the denominator before it was reciprocated, as well as after.

To add and subtract rational expressions we must first determine the lowest common denominator and then rewrite the fractions as one big fraction, using the lowest common denominator. We may only add or subtract the numerators, the common denominator remains the same. Once we have determined our final answer and reduced it, we must state the non-permissible values.

I have included an example below that shows the detailed steps I would take when adding rational expressions and how to state the non-permissible values.

This week in Pre Calc 11 we learned how to graph Reciprocal Value Functions. A function is reciprocated when the values are flipped, for example the reciprocal of is because the values flipped. In order to graph reciprocal linear functions we must first graph the parent function. Next we will be able to find the horizontal and the vertical asymptotes. In grade 11, the horizontal asymptote will always be drawn along the x-axis, which means the equation for our horizontal asymptote is To determine where the vertical asymptote will be, we must find where our parent function intersects the x-axis, which will be our x-intercept and draw a vertical line through it. This line will be our vertical asymptote, represented by and it’s equation is Once we have determined our asymptotes, we now must find the invariant points. The invariant points are are determined by where the parents function crosses the numbers and on the y-axis. We are now able to graph our reciprocal linear functions by starting at the invariant points and drawing two hyperbola’s that follow along the horizontal and vertical asymptotes, gradually getting closer to them, but never actually touching them.

An example of how to graph a reciprocal linear function is

The first step is to graph the parent function which is is has a slope of and it’s y-intercept is

Next we must find the verical and the horizontal asymptotes. Which will be and because as we can see in the graph, the x-intercept of the parent function is Next we must find the invariant points which will be and Finally, we are able to draw our two hyperbola’s.

This week in Pre Calc 11 we learned how to Solve Absolute Value Equations both Graphically and Algebraically. In order to solve an abosulte value equation graphically, we must first seperate the equation into two parts. In the linear equation example: the seperated equations would be: and Next, we would graph both equations in order to visually determine how many possible solutions the absolute value equation might have and the vaules of the possible solutions. Linear absolute value equations can have 0, 1 or 2 solutions. We must also remember that since the equation has abosulte value bars around it, the equation can not be graphed in the negative zone of the graph and will instead reflect back up into the positive zone, making a V shape.

Once we have graphed both equations, we are now able to see that the equations intercept at and This also shows us that this absolute value equation has two solutions.

It is also possible to have a quadratic abosulte value equation, which follows similiar steps to that of a linear abosulte value equation. In the example: we must seperate the equations to be able to graph them. The sperated equations are: and Next we graph both equations to determine their points of intersection. Quatratic absolute value equations can have anywhere from 0 to 4 solutions.

Once we have graphed both equations, we are now able to see that the equations intercept at and This also shows us that this absoulte vlue equation has three solutions.

This week we also learned how to solve absolute value equations algebraically.  In the example: we must first start by getting rid of the absolute value bars by putting this equation into two different equations using piecewise notation. This means that our two new equations will be and Once we have our new equations we must solve to determine the value for which will tell us how many times the eqautions will intercept. In the first equation:  we must first make the equation equal to zero by moving to the other side which gives us from there you continue to isolate and in the end you will determine that In the second equation:  we must follow similar steps, however we must first distribute the negative into the brakets before we can make the equation equal to zero. Once you isolate for you will determine that This tells us that the equation  will have two solutions. Once you have determined the values for you must verify that they are not extraneous answers by plugging them back into the original equation.

This week in Pre Calc 11 we learned how to graph linear inequalities in two variables. We recalled that a linear inequality is different from a quadratic inequality because linear means that it only has a degree of one. We also learned how to identify the y-intercept and the slope of a linear inequality. For example in the inequality: the y-intercept is the second term which is positive and the slope is the first term which has a rise of and a run of We also learned how to determine whether the boundry line of the linear inequality will be broken or solid. If the linear inequality is greater than, less than or equal to (includes), then the boundry line will be solid, however if it is only greater than or less than (excludes), then it will be broken. For example in the inequality: the inequality sign tells us that the value for must be greater than to make this inequality a true statement, this means that the boundry line for this inequality will be broken.

Once we learned how to determine the y-intercept, the slope and whether the boundry line will be broken or solid, we began to graph. In the first example we used, we know that the slope is the y-intercept is and that the boundry line will be broken. If we were to draw this linear inequality on a graph it would look like this:

Once we have graphed the linear inequality, we must determine the possible solutions that will satisfy the inequality by shading the one side of the graph. To determine which region will satisfy the inequality we choose points (called the test points) on either side of the boundry line and input them into the linear inequality. If the inequality sign is true to the numbers then shade in the region of those points, if not shade in the opposite region of the graph.

I have included an example below that shows the detailed steps I would take to determine which region on a graph to shade to find True possible solutions.

This week in Pre Calc 11, we spent most of the week studying and reviewing for the upcoming midterm. I decided to focus my studying on the first unit we learned this year which was Arithmetic and Geometric Sequences and Series. I found that I needed to review the differences between them and also how to calcualte the value of the last number in an Arithmetic Sequence when the number of terms is given, as I have found these things hard to remember.

While studying, I have recalled that the difference between an Arithmetic and a Geometric Sequence is that an Arithmetic Sequence, increases or decreases by a common difference. Where as a Geometric Sequence increases or decreases by a common ratio.

I also reviewed how to find the value of the last number in an Arithmetic Sequence, when the number of terms is given. For example, in the sequence:  7+11+15+19… we need to determine the value for term 12. We already know that is our first term and that this sequence has a common difference of which means that in order to determine tha value of term 12 we will need to input the information that we already know into the equation = as I have shown in the example below.

After solving the equation, you now know how to calculate any term in the sequence that might be asked. This week I also reviewed the differences of how you can tell if a Geometric Series is diverging or converging. To determine if it is diverging or converging, you look at the value of the common ration of the Geometric Series. If it is diverging, it means that or that If it is converging, it means that or that If the Infinite Geometric Series is diverging it means that it will have no sum, however if it is converging than you are able to caculate the sum.

Each of the things that I have mentioned have all been things that I focused my review on over the past two weeks. They were all units that I needed to review as I had forgotten the majority of it.

This week in Pre Calc 11 we learned about equivalent forms of Quadratic Equations. Each Quadratic Function can be written in three different ways: Vertex Form, Factored Form and General Form and each form reveals different information that you may need to be able to graph a function.

A quadratic function in Vertex form tells us the scale factor and the coordinates of the vertex It looks like this:

A quadratic function in Factored form tells us the scale factor and the x-intercepts. It looks like this:

A quadratic function in General form tells us the scale factor the y-intercept and it also tells us if the parabola opens up or down. It looks like this:

We also learned how to find the vertex of a Quadratic function when it is in General Form. To do this you must first complete the square to solve for the vertex, because there are two variables in the function it will not be possible to solve completely, however it will give you the coordinates of the vertex in order to later graph it. Ex. In this example you must first get rid of the coefficient by dividing the first two terms by three. Ex. Then you must find the zero pairs by dividing the middle term by two and then squaring it. Ex.  Then you must simplify the first three terms and multiply the coefficient by to remove it from the brackets. Ex.  You are then able to combine like terms. Ex.  Once you have changed the equation into Vertex Form, you will be able to find the vertex needed to graph it. Vertex

If you wanted to find the x-intercepts, you would have to change the function from General Form to Factored Form by expanding the function and then factoring it.

This week in Pre Calc 11 we learned about properties of Quadratic Functions. We learned how to determine the vertex of a Quadratic function and if it is minimum or maximum, the line of symmetry, the x and y intercepts, the domain and range and we also learned how to determine if a function is going to be congruent to the parent function: Using all of these properties that we learned, we were then able to analyse Quadratic Functions written in Standard Form or as Mrs. Burton likes to call it Vertex Form, to later be able to graph them very easily. This is an example of a Quadratic Equation written in Vertex Form: This Vertex Form is also known as  Knowing what each of the different variables represents in this form can help you analyse the function and allows you to get a really good sense of what it would look like on a graph.

In Vertex Form, :

is the vertical translation, which means that it will translate either a certain number of units up or down along the y axis. It also tells us what the in the vertex is going to be.

is the horizontal translation, which means that it will translate either a certain number of units to the right or to the left, moving along the x axis. It also tells us the line of symmetry and what the in the vertex is going to be. However we must remember that the sign changes when you take the  value out of Vertex Form and put it into your vertex. Example:  the vertex for this function would be 

tells us the parabola is going to be congruent to the parent function, if it going to be a reflection or not and if it is going to be a stretch or a compression. If the value of is is a number other than 1, then the parabola will not be congruent to the parent function. If the sign is negative the parabola will have a maximum vertex, which means that it opens down and that it will be a reflection, however if positive the parabola have a minimum vertex which means that it will open up. If the value for  is a fraction then the parabola will become wider and will be a compression and if the value is a whole number, then the parabola will become skinnier and will be a stretch. Example:  in this function, the parabola will be a reflection, it will have a maximum vertex which means that it will open down and it is a stretch.

In the example that I used throughout this post, we discovered that the vertex is going to be  that it is not congruent, that it is a reflection and has a maximum vertex. If I were to graph this function with the information that I have determined, the function would look like this:

One thing that I learned this week in Pre Calc 11 is how to find the discriminant of a quadratic equation. The discriminant does not solve a quadratic equation, however it can help you avoid attempting to solve an equation with no solutions and it can also help determine the nature of the roots. The formula used to find the discriminant is also part of the quadratic fromula, There are three possible outcomes when you calculate the discriminant, it will either show you that the equation will have one solution, two solutions or no solutions at all. The discriminant is a really easy and fast way to check to determine whether or not it is worth taking the time to solve an entire equation.

If the discriminant is a number that is greater than zero then it will have two possible solutions and it also means that if you were to graph the equation it would intercept the axis two times. For example, if the discriminant is  than it is a distinct root, with two solutions, it is a rational root, because it is a perfect square and it is also a real root because it is greater than zero.

If the discriminant is a number that is equal to zero then it will have one possible solution and it will only intercept the axis one time on a graph. For example, if the discriminant is  than it is an equal root, with one solution, it is an irrational root and it is also a real root because it is greater than or equal to zero.

If the discriminant is a number that is less than zero then it will have no possible solutions and it will not intercept the axis at all. For example, if the discriminant is  than it is not a real root because it is less than zero (a negative number) and it will have no solutions.

I have included an example below that shows the detailed steps I would take to find the discriminant of a quadratic equation.