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As the input gets larger the output gets smaller and approaches 0.

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As the input increases the output also increases. This will go on to infinity. The y value is the square root of the x value. Because it is impossible to find the negative square root all x values must be positive.

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f(x)=a (sqrt b(x+c))+d

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Domain: [0,∞)
Range: [0,∞)
End Behavior:
As x → ∞, y → ∞

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Verbal

Algebraic
y=ax^2+bx+c
Numeric

Graphic

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Verbal

Algebraic
y=mx+b
Numeric

Graphical

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Armando Romero - Exponential
Connor Davidge - Rational
Henry Robertson - Radical
Harrison Limbaugh - Linear & Quadratic

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f(x) = p(x)/q(x)

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Verbal
The input is the same as the output.
Algebraic
y=mx+b
Numerical
(see below)
Graphic

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As the input gets larger, the output gets larger and approaches ∞.

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This is the basic equation:
f(x)=a (sqrt b(x+c))+d

a creates a vertical stretch 2(sqrtx)
b creates a horizontal stretch (sqrt2x)
c creates a horizontal shift  (sqrtx+1)
d creates a vertical shift sqrtx)+1

The inverse:  The inverse is represented by x^2 . The inverse is also ((x-d)a)^2/b-c

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f(x) = 1/x

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f(x) = 1/x

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f(x)=ab^x

b:
- constant/change factor
- always positive
- value raised to the x
a:
- starting/initial value/y-intercept
x:
- independent variable/input of the function

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f(x)=3^x

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Domain: (-∞, 0) (0, ∞)
Range: (-∞, 0) (0, ∞)
End Behavior:
- As x → ∞, y → 0
- As x → -∞, y → 0
Common Transformations:
- Translations → y = 1/x +1
- Compressions → y = 0.5/x
- Stretches → y = 2/x
Inverse: For the parent function of  f(x) = 1/x the inverse is 1/x. However for more complicated functions the inverse may not be a function.

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f(x)=3^x

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- The importance of "a" in an exponential function is that, exponential functions will always increase or decrease so they must have a starting value to know where to start decreasing or increasing.
- "b" is important in exponential functions because it's the value at which the function will either increase or decrease. Some may see this as the percentage that an equation changes by. "b" will always

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Verbal
y is porportional to the square of x
Algebraic
y=ax^2+bx+c
Graphic

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Domain: (- ∞, ∞)
Range: (0,  ∞)
End Behavior:
- As x→ ∞, y→ ∞
- As x→ -∞, y→ 0
Common Transformations:
- Translations→ y=3^(x)+2
- Compressions→ y=0.5(3)^x
- Stretches→ y=2(3)^x
- Inverse→ x=3^y

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Rational functions can have holes which are single points in the graph that cannot exist. Holes are created when the input causes both the numerator and denominator to be equal to zero.

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If a rational function has the same factor on both the top and bottom, the graph will show a hole.
e.g. f(x) = (x+5)(x-3)/(x-3)

If you set the denominator to 0 you can find the vertical asymptotes
e.g. f(x) = 3x - 1/x - 5
x - 5 = 0
vertical asymptote is at 5

If the variable numerator and the denominator are both to the same degree, then divide the coefficients.
e.g. f(x) = 6x^2/2x^2
= 6/2
Horizontal asymptote at y = 3

If the number in the numerator is a lower degree than the denominator, y = 0 is the horizontal asymptote.

If the number in the numerator has a higher degree than number in the denominator then there is no horizontal asymptote.

(Source)

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Rational Equations can be used to model multi-person work problems

Person 1 lays 500 bricks an hour
Person 2 lays 400 bricks an hour

How long working together will it take them to lay 1800 bricks?

By using two rational equations you can create a proportion allowing you solve for the unknown amount of time. In this example it is two hours

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- Exponential functions are some of the most commonly used type of functions in real world scenarios. Populations are shown with exponential functions, investments are computed using exponential functions, and exponential functions are even used to carbon date artifacts (AlgebraLAB).

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Domain (-∞,∞)
Range (-∞,∞)
End Behavior
As x →∞, y→∞
As x→-∞, y→-∞

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Linear can be applied to a lot of different things such as the distance you travel by car. X is distance and Y is time and it will go in a straight, linear fashion.

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Quadratic functions can be applied to the arch of a ball thrown through the air. The ball will arch resembling the quadratic function.

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The most important part of a radical function is the starting point of the function.
for the simple equation of sqrt of x the graph will start on the orgin and move to the right. "c" and "d" affect where the function starts "a and "b" determine the direction of the swoosh.

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Linear function's slopes are shown in their equation. The slope is the number that directly modifies the x.

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Domain: (-∞,∞)
Range: [0,∞)
End Behavior:
As x approaches ∞ Y approaches ∞

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Quadratic Funtions

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Suppose $4000 is invested at a 6% interest rate compounded annually. How much money will there be in the bank at the end of five years? At the end of 20 years?

Assigned variables:

x= time in years and

y= amount of money in investment account.

y=4000⋅(1.06)^x

To find total amount of money after 20 years:

y=4000⋅(1.06)^20

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There are many different types of radical functions. x under a cube root sign is also a radical function. This continues on forever. Because of this radical functions can look very different for example, cube root radicals go on forever in both directions, there is no starting point.

This means the final basic equation is a(b(x-c)^1/e-d. I didn't include this in earlier entries for simplicity.

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Radical functions are found everywhere. Rocket scientists and the simple Joes both use radical functions. This is because our world is built upon exponents and thusly radicals.

For example you have a square garden that has an area of 25 feet. How much fence in feet do you need to wrap the entire garden. The first step of this problem is to take the square root of 25 which is 5. Then you times 5 by 4 to get 20 feet of fence.

Radical functions are also used in problems involving volume, gravity, and even plumbing.