This week in Precalc 11, I learnt how to graph inequality problems involving two variables. I decided to write about it because I find it difficult to do and I think that by going over it, I will have a greater understanding of the subject.

The first step to graphing the inequality is knowing the information given to you as the inequality could be written in quadratic or linear form. In this example the inequality is written in linear form and from this we can see that the run and rise is 2, and the y intercept is -3. We can then graph the line. When we have a greater than or less than symbol, we have to shade one side of the line to show that it is the side with possible solutions. To determine which side to shade you can plug in a coordinate into the inequality equation and if it is true, then you shade that side of the line and if it is not, shade the other side. If I plug in (0,0) to the inequality, 0 is not smaller or equal to -3 so this means that I shade the opposite side. The inequality sign also determines if the line of the equation is solid or dashed. It is solid in this example because the sign is equal to, if it is not equal to the line is dashed.

In this example, the inequality is written in quadratic form. First we can determine that the vertex will be placed at (0,-3) and have a stretch value of -5. Graph the parabola. To determine what to shade, it depends on which way the inequality sign faces the Y. For this example, since the inequality sign is indicating y is greater than and the parabola is opening up, you will shade the inside of the parabola because that is where the value of y is highest. You can also use the method above of plugging in a coordinate to determine where to shade. The line is dashed because the inequality sign is not equal to.

hope this helps:)

https://www.desmos.com/

This week in precalc 11, I learnt how to graph and solve one-variable inequalities. I chose this subject because it serves as the foundation for understanding how to draw more complicated inequality graphs, such as those involving two variables.

In this first example, we don’t have to solve the inequality, but we have to graph it. The type of dot you use depends on the greater than or less than sign. If the greater than or less than sign has a line under it, you will use the solid dot as it indicates that x can also be the value, if the sign does not have a line, use a hollow dot as it indicates the value is not included. In this example we will use a solid dot. Place the solid dot on the value of x, 3. To determine where the arrow will point it depends if it is a greater than or less than sign in the inequality. When x is on the left side of the inequality, the sign pointing to the right like an arrow, is greater than, but if it is pointing towards x, left, it is less than. In this case the arrow will point to the left.

 

For this example, the first thing you want to do is solve the inequality, you do this by isolating the variable x. Something you have to remember when solving inequalities is that when you divide by a negative, you must flip the greater than or less than sign. This is because dividing by a negative makes the greater than or less than sign no longer true to the equation. After isolating the variable, you can then graph the inequality. Figure out the type of dot you will use as explained above and graph it. Next, determine where the arrow will point, in this case it will point to the right.

hope this helps:)

How do you find the transformation of a function?

For a function y = f(x), if a number is being added or subtracted inside the bracket then it is a horizontal translation. If the number is negative then the horizontal transformation is happening to the right side. If the number is positive then the horizontal transformation is happening to the left side.

Definitions

vertex –  is a point where two straight lines meet.

conjugate – A math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of x + y is x – y.

maximum – a point at which a function’s value is greatest. If the value is greater than or equal to all other function values, it is an absolute maximum.

minimum – The smallest value of a set, function, etc.

domain – is the x values in a function such as f(x)

range – is the y values in a function such as y =.

Function Transformations. Transformation of functions means that the curve representing the graph either “moves to left/right/up/down” or “it expands or compresses” or “it reflects”. For example, the graph of the function f(x) = x² + 3 is obtained by just moving the graph of g(x) = x² by 3 units up.

We can tell this when looking at our equation:

 

Hope this helps 🙂

 

This week in PreCalc 11, we started to learn about quadratic functions.

The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in “width” or “steepness”, but they all have the same basic “U” shape.

Definitions:

The Maximum value of a function is the place where a function reaches its highest point, or vertex, on a graph.

The minimum value of a parabola is the y-coordinate of the vertex of a parabola that opens up

In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

 

 

The parent function of the quadratic family is f(x) = x2. A transformation of the graph of the parent function is represented by the function g(x) = a(x − h)2 + k, where a ≠ 0.

This week, I gained knowledge on the application and timing of the quadratic formula. Because factoring or completing the square may not always be an option when solving a quadratic equation, it is crucial to know how to employ the quadratic formula. When factoring and then completing the square doesn’t work, you can fall back on the quadratic formula. The numbers from your equation are used to replace the three variables in the quadratic formula, a is replaced by the leading coefficient, the middle coefficient is b and the constant is c.

See examples of using the formula to solve a variety of equations:

In this example the trinomial could have been solved by just using factoring techniques. The quadratic formula is more commonly used for questions that cannot be factored to whole numbers.

Usually when using the quadratic formula you will have two solutions but in some cases you might have one or none. To know early on without fully solving the formula to see how many solutions you will have, you can look at the discriminant (radicand). If the discriminant if a weird number that you don’t see often like 0, you will have one solution, imperfect and perfect squares have two solutions, and numbers that break rules within math like a negative in the radicand of a square root have no solutions.

√0 ← one solution     √49 or √12 ← two solutions     √-5 ← no possible solution

hope this helps 🙂

 

This week in math 11, I learnt how to solve radical equations using factoring.

An equation in which a variable is in the radicand of a radical expression is called a radical equation.  To solve a radical equation: Isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them. let’s get into it!

As usual, when solving these equations, just like we learnt in grade 10. what we do to one side of an equation we must do to the other side as well. Once we isolate the radical, our strategy will be to raise both sides of the equation to the power of the index. This will eliminate the radical.

Step one is to isolate the radical on onside of the equation, in this case we move positive 4 over to the other side of the equation making the 4 negative and it nows reads = x – 4. our next step is to raise both sides of the equation to the power of the index. Since the index of a square root is 2, we square both sides. by doing this we now have a new equation with not radical. Solve the new equation, Remembering that (a)² = a. On the other side of the equal sign, since we have (x – 4)² we double the brakes and FOIL the equation. This will leave us with a trinomial on one side and a binomial on the other. Next we need to combine the binomial into the trinomial and have 0 on the other side. since we don’t want a negative x² we will subtract x and add 2 into the trinomial.

our send part is too factor our equation then solve for x, in this example I got two possibilities.

our last part is to verify our answer. 1st I tried x = 3, and by inputing 3 into x’s spot I found that x is equal to 5 and it does not match my answers, so the only possibility is x + 6.

hope this helps  🙂

 

 

This week in math 11, I relearned how to correctly factor trinomials. Learning how to factor expressions is important because it helps you answer your solution making it simpler to read and verify it so you know its correct.

A Trinomial is a polynomial that contains three terms. The first term is an x² term, the second is an x term, and the third is a constant (a single number).

ab+a+b     or      x²+x+12

To factor a trinomial, you need to find two numbers that can multiply the constant while also adding/subtracting from the coefficient of the variable with no apparent exponent. Once you have those two numbers, split them using brackets. This will also include the variable from the original expression, resulting in the same variable but different numbers in both brackets. To determine the signs that separate the numbers from the variable in each bracket, consider that when multiplying the numbers together, the sign should match the one on the constant, and when adding or subtracting the numbers, the sign and number of the coefficient with no visible exponent should match.

EXAMPLE 1)

 

In the example above, we have x squared, minus 7x, plus 12. There are three combinations can multiply to 12 including 12 x 1, 3 x 4, and 2 x 6. We can limit our options by deciding which of these can add or subtract to make the negative coefficient of 7. 3 plus 4 equals to 7 so that is the combination we are going to use. but we have to do the opposite integer to get negative 7. We are then going to create two brackets containing the variable, in this example x, and put one in each bracket and separate the combination of numbers we just decided on, – 3 and – 4, and put them in the same brackets as the variables. To determine the sign that will separate the constants from the variables, you want to think about the signs attached to the coefficient and constant in the original expression. Since the constant is 12 and the coefficient of x is -7. The signs both have to be negative so they can add up to -7 and multiply to become a positive to equal +12. To check you factored correctly, you can foil, expand it, or box method, and compare to the original expression, which should result in being the same.

EXAMPLE 2)

This example follows the exact same steps as example 1. This example just has exponents on two out of the three terms. When doing this question you want your brackets to have exponents that match the middle value.

EXAMPLE 3)

 

This example follows the same steps and example 1 and 2. But in this example there is a coefficient attached to the first term. To get ride of it for the time being you have to multiple the 3 into the third term to make it go away giving us 18. Then you continue with your normal steps finding number that both can be multiplied to 18 and added to 7. once you have your correct brackets there is one more step. you then divide your final numbers by the 3 from earlier. as you can see above the -9 can be divided by three giving us -3, but the two can’t be divided to get a whole number. so the three gets placed in-front of the t³. Giving us our final answer.

hope this helps 🙂