Archive of ‘Grade 11’ category
In this week of precalc 11, we learned how to solve rational expressions, which are fraction questions with variables. One thing in specific we learned about was non-permissible values. These are values that x cannot be in each equation. First, here are some examples of what rational expressions can look like:
These expressions cannot have square roots, or exponents that are variables.
A non-permissible value is also known as a restriction, the domain, or values that the expression cannot be defined by. This means there are certain numbers, that when put in place of x, make the denominator zero. When the numerator is zero the equation can still be simplified. However, when the bottom of the fraction (the denominator) is zero, this is not possible. Therefore, these are non-permissible. When you write these restrictions, use the equal symbol with a line through it, meaning “x cannot be equal to…”
Here are the non-permissible values for the expressions shown above:
For example, in the second expression, x cannot be -12. If it was, it would look like this:
Since the denominator is now zero, which is impossible, it means -12 is the restriction on this expression. There can also be multiple restrictions, depending on the fraction and number of variables. The last example had a fraction as a restriction, which is not as easy to solve in your head. This is how I solved to find -3/4:
Here are a couple more examples:
In this week of precalc 11, we learned about solving inequalities. This was partly a review from grade 9, with new skills added on. Linear equations make a line on a graph, while quadratic equations create a parabola (a curved symetrical shape). Both of these equations can be inequalities as well.
Here are some examples of linear inequalities:
6x > 18
4x + 1 < 9
3x – 3 ≥ 4x + 1
These are all quadratic inequalities:
x² + 3x + 2 ≤ 0
2x² > -3x – 7
x² – 7x + 9 ≥ 9
To solve an inequality, you must get the variable on its own. For example:
For this inequality: 6x > 18, you divide both sides by 6, so the x is by itself. Therefore, x would be greater than 3 (x > 3)
To solve this inequality: 3x – 3 ≥ 4x + 1, start by subtracting the 4x from both sides. Then add the 3 from the left side to the right. This would create the equation: – 1x ≥ 4. To make x positive, divide both sides by negative 1. This would make x less than or equal to 4 (x ≤ -4), because when you multiply or divide by a negative, the sign changes.
Here is one final example using the quadratic inequality: x² – 4x > 5
This is done a bit differently than the linear inequalities because you need to factor. First, move the 5 to the left side so the inequality has the zero on the right: x² – 4x – 5 > 0
Then, factor the right side: (x – 5) (x + 1) > 0
Next, find the roots. This is the two numbers, 5 and 1, but with opposite signs: +5, -1.
Finally, write out the solution: x < -1, and x > 5. This is the answer because the original question asked where the inequality was greater than 0. Because this equation would open up and have a minimum vertex (the x² was positive), it is greater than zero to the left (less than -1) and to the right (greater than 5) of the parabola. In between the parabola, x is less than 0, so it wouldn’t satisfy the inequality.
In week 10 of my precalc 11 class, we learned about three different forms of equations that can be graphed, and how to convert them from one to another. Here are the different forms (with examples below):
General Form: y = ax² + bx + c
y = x² – 2x + 4
y = -2x² + 5x -16
Standard or Vertex Form: y = a(x – p)² + q
y = (x – 4)² + 7
y = 3(x – 2)² – 1
Factored Form: y = a(x – x₁)(x – x₂)
y = 2(x + 7)(x – 3)
y = (x – 5)(x – 2)
Each of these equation types can show different things about what the graph will look like. For example, the last number (with no variable) in general form tells you the y-intercept. The 2 numbers after the x in factored form are the roots of the equation (make sure to change the signs first). In order to fully graph a quadratic equation, sometimes you need to convert one form of the equation into another. Here is how to convert general form into factored or standard:
This is how to change standard/vertex form into general or factored:
Finally, here is how to convert factored form into general or standard:
This week in precalc 11, I continued learning about the connection between graphing and quadratic equations. One thing specifically that I learned about was how to plot the vertex on a graph, when given a quadratic equation. The “parent function” is like the original graph. It is created from the equation y=x². For this graph, the vertex is at the coordinates (0,0). This is what the parent function graph looks like:
When graphing an equation different than y=x², there are a few ways to find the coordinates of the vertex. First, see if there is a last number (without a variable) in the equation. This tells you where the vertex will be on the y-axis. If the number is positive, move the vertex upwards (the amount of spaces the number says). If it is negative, move down. As you can see below, the parabola is still the same shape and size as the parent function, just shifted in a different direction. Here are some examples:
y = x² + 2
y = x² – 7
If there is a number either added or subtracted from the x, this tells you where the vertex is on the x-axis. If the number is positive, move the vertex to the left. If it is negative, move to the right. Here are some examples:
y = (x + 1)²
y = (x – 3)²
Finally, these two things can be combined in an equation, so the vertex has coordinates that do not include zero. Here are a few more examples:
y = (x – 3)² + 4
y = (x – 1)² + 1
In my precalc 11 this week, we learned about the quadratic formula. This is another way to solve quadratic equations, other than factoring or completing the square. The quadratic formula looks like this:
It is easy to solve an equation using the quadratic formula because all you do is put in the numbers, and use algebra to find the variable. To do this, you must find out what numbers in the polynomial are A, B and C. A is always the number with the x², B is always with the x, and C is always the last number, without a variable. Remember that when there is no number with an x, there is an invisible 1. Here are some examples of what A, B, and C are in the following polynomials.
This is how to solve a polynomial using the quadratic formula:
In the next equation, the quadratic formula is used, but the answers are 2 rational roots. This shows that the equation could have been factored from the start, rather than doing the quadratic formula, which sometimes takes longer. When you need to solve a quadratic equation, first check if the numbers can be factored, or if you can use the technique of completing the square. If not, using the quadratic formula will guarantee the correct answer.
In my precalc 11 class this week, I learned about graphing, and how it can relate to a quadratic equation or function. A quadratic function is in the form y=ax² + bx + c
These are all quadratic equations: Even if they don’t look like it at the start, they can be moved around into the correct form.
These graphs in curved shapes are called parabolas:
When an equation creates a graph in a parabola, it is a quadratic function. Parabolas have a vertex, which is either the highest or lowest point. If it is the highest point on the graph, it is called the maximum point. If it is at the bottom, it is the minimum point. Here are some examples:
This week in precalc 11, I learned how to factor a polynomial, specifically when there is a fraction within it. Here are some examples of polynomials with fractions, that can be factored:
To be able to factor these easily, there should be only integers with the variables. This means you must find a greatest common denominator (GCD), so all the terms have the same number at the bottom of the fraction. Then, the fraction can be removed as a greatest common factor (GCF), and the polynomial can be factored. Here are the above examples, with the steps to take to factor:
Remember, if there is a variable by itself without a number in front, there is always an invisible 1. Also, don’t forget that all numbers/variables are fractions, even if they aren’t shown that way at first. If a variable, like x, is not written as a fraction, it has a denominator of 1.You can also verify your answer when you are done, to check if it is correct. To do this, just take the final answer and use the distributive property with FOIL to get back to the polynomial you started with.
In week 5 of precalc 11, we started learning about factoring. This is a continuation of grade 10, but new factoring skills are being added on. One concept we learned about is factoring with a greatest common factor (GCF). This is when you remove a common factor from all the terms, so it becomes easier to factor. The GCF has to be something that is present in ALL terms. Remember, a variable, like x, can also be a part of the common factor. Here are some examples of binomials with a GCF:
When you remove the GCF from a binomial or trinomial to factor, make sure to remove the same GCF from all terms. If there is a variable like x in two of the terms but not the other, then it cannot be part of the GCF that gets removed. For example:
Finally, here are some more examples of how to remove the GCF from a binomial or trinomial. When you do this, the GCF is written in front, and the remaining the numbers are inside brackets. Sometimes this is all you do to simplify the question, but make sure to check if you can factor any further.
In my precalc 11 class this week, I learned how to rationalize a denominator. This happens when there is a fraction with a square root on the bottom. For example:
When fractions are being simplified, it is better to show them with an integer on the bottom. If the denominator is not an integer, but a radical, you can rationalize it. To make the denominator a rational number, or a non-radical, multiply the whole fraction by 1. This is done my multiplying it by the radical on the bottom, that you want to get rid of. (Since the radical is on the top and bottom of a fraction it equals 1).
These fractions below are all equal to 1, because the numerator and denominator are the same:
When you multiply the two fractions, make sure to multiply the radicals only with other radicals, and integers in front of the radicals (if it’s a mixed radical) with other integers. If there is no number in front of a radical, pretend there is a 1. Make sure not to multiply a radical with an integer! Here are a few examples of how to rationalize the denominator:
In precalc 11 class this week, we learned about adding and subtracting radical expressions. This involves combining like terms that are radicals. Like terms is something I had learned about before, but I didn’t realize it was possible with radicals. Terms are numbers, variables, and/or exponents in groups. Here are four examples of terms:
When the variables and exponents in more than one term are the same, the terms are “like”. The coefficients (number in front of the variable) can be different. For example:
When multiple radicands and their index are the same, they can be combined. For example, these radicals are like terms, because they have the same radicand and index (the square root of 4):
It is important to remember that the coefficient can be different for each term, because that is what’s being combined. To combine like terms, add/subtract the coefficients based on the sign in front of them. If there is no sign, it is positive. Here are the steps to combine like terms that are radicals:
If the radical has no number in front, there is an invisible coefficient of 1. Don’t forget to add or subtract it as well! Here is one final example: