The parent function of a parabola is y = x²

Vertical translation is shown when there is a term added to x². So, if y = x² + 5, the parent function would be moved 5 units up.

If y = x² – 7, the parent function would be moved down 7 units.

Horizontal translations are shown when y = (x – p)², p being the translation. If the parent function moves 5 to the right, we replace p with 5, y = (x – (5))²

If p = -4, we replace p and the graph moves 4 to the left, y = (x – (-4))² or y = (x +4)²

Now, if we add this together, we can understand how a parent function changes.

y = (x +4)² – 5

This means that the horizontal translation is 4 units to the left, then 5 units down.

The scale can affect whether the parabola is stretched or compressed. If there are no terms before the brackets, the scale is as original, 1, 3, 5, etc. If there is a number higher than 0 in front of the brackets, y = 2(x +4)² – 5, the parabola becomes stretched.

If the number is less than 1 and greater than 0, y = .2(x +4)² – 5, the parabola is compressed.

Lastly, if this term before the brackets is negative, the parabola becomes down facing, y = -(x +4)² – 5

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A quadratic equation is an equation that has an x² and is equal to 0. On a graph, the lines will look like a v. This is because, when solved, this equation will have two answers. There are 3 ways to solve a quadratic equation, I’ll be showing how to use one of them, factoring.

Now, we have a couple of questions we must ask ourselves when factoring.

Is there anything in common? Remove it.

15x³ – 5x²

5x²(3x – 1)

Is it a difference of squares?

There is a difference of 3x and 1, but they are not both squares.

25x²-49 would be a difference of squares that could then be factored.

25x²-49

(5x – 7)(5x + 7)

If we are still left with factorable terms, and there are three terms left, is there a pattern? Is it easy to factor or hard?

The pattern we look for is ax² + bx + c

If a = 1, it is easy to factor and we just need two terms that have the product of c and the sum of b.

x² + 7x + 10

The factors of ten are : 1 x 10, 2 x 5

2 and 5 have the sum of 7

Both terms are positive so the factor is then (x + 2)(x + 5)

If a is a number higher than 1, we can use methods such as guess and test.

5x² + 9x + 4

Since the sum of 5 and 4 is 9, we can guess that the factor is

(5x + 4)(x + 1)

Because we distributed, it equals 5x² + 9x + 4

 

Now, we can use what we know of factoring terms to solve a quadratic equation.

A quadratic equation looks like

s² – 2s – 35 = 0

There is an x² term, and it is equal to 0.

 

To solve this, we can factor one side. Since there is no common factors, we can skip difference of squares (since there is 3 terms, not 2) and factor the pattern the easy way. -7 and 5 have the sum of -2 and -35.

(s – 7)(s + 5) = 0

s – 7 = 0

+ 7

s = 7

s + 5 = 0

-5

s = -5

s equals 7 or -5

When there is an equality sign (=) in an equation, we can further solve it.

For example: √x=10 is a radical equation

We can solve this equation by isolating the variable, x.

√x=10

To rid of the √, we can square both sides.

√x²=10²

x=100

Voila, we have found what x equals. To finish, since there was a variable under a √ sign, we need to show restrictions.

This is done by taking whatever is under the radical with an even index, and saying that x≥0.

Mind you, if the index of the radical wasn’t a multiple of 2, but an odd number like 3, x is an element of the real numbers and has no restrictions.

Then, we should always check to make sure our answer is correct.

√x=10

√100

10 = 10

 

Another example would be the equation √x+3 = 5, where x + 3 is all under the radical sign.

We do the same thing of squaring both sides

(√x+3)² = 5²

x+3 = 25

Isolate the variable by removing -3 from both sides

x = 22

Now, we can find the restriction

x+3≥0

Minus 3

x≥-3

X is greater than -3 so our restriction is correct and we can do a check of the equation

√x+3 = 5

√22+3

√25

5 = 5

This answer works.

 

Some things to keep in mind

Some answers may not work, they are called extraneous solutions and have no solutions. That is why it is important to check the answer.

When finding restrictions, if there is a need to divide a negative term from both sides, the sign switches. This means that if it is 7-6x≥0

We isolate the variable by first -7

-6x≥7

Since we then divide by a negative 6, the sign changes

x≤7⁄6

A radical expression that contains positive terms and negative terms have ways to be further simplified.

When a radical in an expression like such is presented: √4a + √16a – √9a, a ≥ 0

This is a radical expression because there is no equal sign and it includes radicals.

Since this expression has + and – symbols, we are able to combine like terms.

Combining like terms is done when the index of the root (4∛8, 3 is the index) is the same. The coefficient (4∛8, 4 is the coefficient) is the number that changes when the terms are combined and the radicand (4∛8, 8 is the radicand) stays the same.

Lastly, we need an understanding of simplifying radicals to help us simplify terms to create the same radicand and combine them.

√4a can be simplified to 2√a because √4 = 2

√16a can be simplified to 4√a because √16 = 4

– √9a can be simplified to – 3√a because √9 = 3, 3 x – = -3

√4a + √16a – √9a

2√a +  4√a – 3√a

Since the index and the radicands are the same, we are able to combine them to simplify the expression

3√a, a ≥ 0

 

Another example would be: 5e√24e³ – 7√54e⁵ + e²√6e + 6e, e ≥ 0

Each term is first simplified by finding the perfect square inside the radical and removing it. When it is removed from the radical, it is also rooted by the index (2√12
2√4⋅3
2√4⋅√3
2(2)√3
4√3)

5e√24e³ – 7√54e⁵ + e²√6e + 6e, e ≥ 0

5e√4⋅6e²⁻¹ – 7√9⋅6e²⁺²⁺¹ + e²√6e + 6e

5(2)e¹⁺¹√6e – 7(3)√6e²⁺²⁺¹ + e²√6e + 6e

10e²√6e – 21e²√6e + e²√6e + 6e

Then, since the index and the radicand is the same, the terms can be combined and simplified

-10e²√6e + 6e, e ≥ 0

 

The absolute value of a real number is defined as the principal square root of the square of a number.

The absolute value is the number of spaces from a number to 0.

The absolute value symbol is where x = real number

The principal square root is when the square root is given and the answer is positive. For example √25 = 5

So when we look for an absolute value of a number (that number being the one inside the absolute value symbol), we look for the number squared and then square rooted.

For Example : The absolute value of $latex \mid 5 \mid$

= √5²

√25

5

Therefore the absolute value of the number 5 is 5

 

 

This week we learned how to determine in a geometric sequence.

Using the equation = , we can find any term in the geometric sequence.

 

For example: The geometric sequence is 2, 8, 32…

 

We can find r, which is the variable for the ratio, by taking the second term and dividing it by the first.

Therefore is the ratio, or 4

Now we know that or the first term is 2

 

So now we can fill in the equation. Let’s find the 7th term.

= 2

= 2

= 2(4096)

= 8192

 

This equation is very helpful to finding the but it can also be helpful if you need to find another variable and you already know two.

Imagine we already knew and we want to find the first term. Using the equation, we can solve for

8192 =

8192 =

8192 = 4096

Now you divide both sides by 4096

2 =