Applications of the function to real-world situations

Transformations

1. Cubic Function Graph

Basic Cubic Function

Basic Shape

The basic shape of a cubic function has a shape of an ¨s¨. It starts in a quadrant and then it has a turning point, that always changes the quadrant from where it started. The turning point is a tangent,a straight horizontal line that goes through the points (0,0). In the cubic graph the value starts in a negative value and when it reaches the turning point it changes into a postive value. This is clearly shown in the picture.

Basic Equation

In a cubic equation, any function with an x cubed, any value that is plugged into the variables may change how the graph looks. A number may change the direction of the graph, the slope of the curve, the turning point and the intercepts. For example in an equation y = a(b(x – h))3 + k, changing any of the variables a,b,h & k alter the graph. The basic equation for a cubic function is y = x3 In this case the variable is x, so if the value plugged in is negative then y will give a negative point in the graph. The same happens with positive values, if the value plugged in for x is positive then the point y will give a positive number in the graph. This is why the shape of the graph. Also the turning point in the graph is due to the equation. For example if you plug in -1 in the equation then the value for y will be -1. If you plug in 0 for x then the y value will be 0, and finally if you plug in 1 for x then the value for y will give 1. This demonstrates the turning point in the graph, through points: (-1,-1), (0,0), and (1,1).

Domain, Range, Intercepts & Vertices

x (domain)

y (range)

intercepts

x = ℝ

y = ℝ

x-int: (0,1), (0,0) and (0,-1)

y-int: (1,0),(0,0) and (-1,0)

Inverse Cubic Fucntion

Basic Shape

The basic shape of an inverse cubic function has a shape of an extended or larger ¨s¨ than its normal counterpart. It starts in a quadrant (usually in and then it has a turning point that almost always changes the quadrant from where it started). The turning point is a tangent (as well as its normal function) , a straight horizontal line that goes through the points (0,0). In the inverse cubic graph, the values for y start at a negative value, and when it reaches the turning point they change into positive values. This is clearly shown in the picture for the shape and both graph of the inverse cubic function shown before.

As it can be seen, the graph takes on a much more spread out, horizontaly, shape than its normal cubic function counterpart. It also does not feature two key intersction points at

both the -1 and

1 x values for the graph, which in turn only features onne clear intersection at point (0,0) on the graph instead.

Basic Equation

In an inverse cubic equation, or any function with the cubed root of x, will be affected by the variables plugged into it (as its normal counterpart), which ultimately will determine how the graph looks. A number may change the direction of the graph, the slope of the curve, the turning point and the intercepts. For example, in the equation y = a(b(x – h))(1/3) + k, changing any of the variables, such as: a, b, h, or k will alter the graph in multiple ways, whether in length, amplitude, and even intersects.

The basic equation, then, for any inverse cubic function is y = x(1/3). For this very specific, basic example of the equation for an inverse function is only x. The values that can be plugged into the equation, being a cubed root and thus allowing answers ranging from being positive or negative, can be either negative or positive. When plugging in a positive X value into the equation, Y will always give you a ppositive number, and the same rule for negative numbers as well. Zero (0) is the only number here which follows another rule, and for which its cube root will always be zero. . This is why the shape of the graph takes the shape that it primarily has, the stretched out ¨s¨shape.

The turning point in the graph is also due to the equation. For example, if you plug in -1 in the equation then the value for y will be -1. If you plug in 0 for x then the y value will be 0. And, finally, if you plug in 1 for x then the value for y will give 1. This demonstrates the turning point in the graph, through points: (-1,-1), (0,0), and (1,1).

There exist currently no restrictions on the value of x that can go againts the inverse function, or that can give a result which is not measurable.

Key Features

- No minimum or maximum values for y.
- Horizontaly asymptotic.

Domain, Range, Intercepts, and Vertices:

Domain

Range

Intercepts

x= ℝ

x= ℝ

x- intercept: (0,0)

y- intercept: (0,0)

Real- World Applications

- Making Box - Height of Water in Spherical Tank

- The Smallest Distance from a Parabola

- Pumping Water out of Tank

- Equation of State for Real Gases

- Electrical Resistance

- Finding Interest Rate

- Break-Even Points in Economics

- Von Neumann’s Model of an Expanding Economy

- Trisecting the Angle

- After the proof of Feramat’s Last Theorem ,Andrew Wiles, Open Problem for the Next Century

All of these are very clear real- world applications for the Cubic Function and its equation, although some are more complicated than others. IN essence, though, a cubic function would probably be used more often to find the volume or determine the volume quantity of multiple objects. This con not only help in determining those values superficially, but for ther related aspects such as bottle production (amount of volume to fit liquid inside) or in the creation of boxes, which are tools essential to the packaging and transportation of other materials. At simple sight a cubic function might not seem relevant at all, but to actually has a great dela to do with everyday measurements and tasks, and can help these in more ways than we could think of.

Transformations

Let's now talk about how transformations affect a cubic function (shifts, streches and reflections). To understand how the Transformations affect the fucntion itself, we are going to work with the following example. The equation is y = a(b(x – h))^3 + k, so we are going to see how a, b, h and k affect the cubic functions graph.

Let's set the cubic function as f(x) = 1/1*(x – 0)^3 + 0. The original Graph looks like this:

We are going to first change the "k" value of the graph, which is outside the argument of the function. As we make this changes, the graph moves either up or down. Here is an example of the graph if we have a positive number for k:

As we can see, if we have h equal 5, then the whole graph Shifts 5 coordinates upward. The same pattern would happen if we subtract 5, but the grpah instead would move down 5 coordinates.

Now lets explore what happens if we change the "h" value. In this case, the number is within the argument. If we have a positive number for h, then that means we are subtracting within the argument. If we have a negative value for h, than we are adding within the arugment (because a number minus a negative is adding). So, if we add within the argument, the graph "Shifts" to the left. If we subtract within the argument, than the whole parabola moves to the right. Here is an example of this phenomena:

As we can appreciate, if we subtract 5 inside the argument, the whole parabola moves 5 coordinates to the right. Its kind of tricky to think of it this way, that adding means it moves to the left and subtracting means it moves to the right. But it makes sense because, if we want to add when we have a subtracting symbol we plug in a negative number. If we want to subtract when we have a subtracting symbol, then we plug in a positive number. Hence, if H is positive, the parabola moves to the left. If h is a negative, then the parabola moves to the right.

The next type of transformations we can deal with are "reflections" and "stretches". If we change the value of "a" of the function, the parabola tends to open or close tightly together. For instance, if we have a be equal a negative number, the smaller the negative number is the closer together it looks. Here is a graph of when "a" equals a negative number:

As we can see, the graph got closer together. Now, another important concept of changing "a" is that, if you change the sign of "a", the parabola will be reflected and the direction of the parabola would be opposite.

Finally, we have "b", which is a constant which is multiplied within the argument of the function. This constant also detemrines how strecthed the graph looks. The bigger, and positive the value for "b" is, the more the graph stretches. Here is an example:

As it can be appreciated, the parabola is more opened than the orginial one. This type of transformation is called a "stretch", because it causes the graph get wider, or thinner.

From this oobservations, we can conclude general rules for transformations. This rules are what causes the different changes in the graph of the cubic (and other) fucntions. The rules are:

f(x) + a is f(x) shifted upward a units

f(x) – a is f(x) shifted downward a units

f(x + a) is f(x) shifted left a units

f(x – a) is f(x) shifted right a units

–f(x) is f(x) flipped upside down ("reflected about the x-axis")

f(–x) is the mirror of f(x) ("reflected about the y-axis")

## Cubic Functions

## Basic Cubic Function

## Basic Shape

The basic shape of a cubic function has a shape of an ¨s¨. It starts in a quadrant and then it has a turning point, that always changes the quadrant from where it started. The turning point is a tangent,a straight horizontal line that goes through the points (0,0). In the cubic graph the value starts in a negative value and when it reaches the turning point it changes into a postive value. This is clearly shown in the picture.## Basic Equation

In a cubic equation, any function with an x cubed, any value that is plugged into the variables may change how the graph looks. A number may change the direction of the graph, the slope of the curve, the turning point and the intercepts. For example in an equation y = a(b(x – h))3 + k, changing any of the variables a,b,h & k alter the graph.The basic equation for a cubic function is y = x3

In this case the variable is x, so if the value plugged in is negative then y will give a negative point in the graph. The same happens with positive values, if the value plugged in for x is positive then the point y will give a positive number in the graph. This is why the shape of the graph. Also the turning point in the graph is due to the equation. For example if you plug in -1 in the equation then the value for y will be -1. If you plug in 0 for x then the y value will be 0, and finally if you plug in 1 for x then the value for y will give 1. This demonstrates the turning point in the graph, through points: (-1,-1), (0,0), and (1,1).

## Domain, Range, Intercepts & Vertices

y-int: (1,0),(0,0) and (-1,0)

Inverse Cubic Fucntion

The basic shape of an inverse cubic function has a shape of an extended or larger ¨s¨ than its normal counterpart. It starts in a quadrant (usually in and then it has a turning point that almost always changes the quadrant from where it started). The turning point is a tangent (as well as its normal function) , a straight horizontal line that goes through the points (0,0). In the inverse cubic graph, the values for y start at a negative value, and when it reaches the turning point they change into positive values. This is clearly shown in the picture for the shape and both graph of the inverse cubic function shown before.Basic Shape

As it can be seen, the graph takes on a much more spread out, horizontaly, shape than its normal cubic function counterpart. It also does not feature two key intersction points at

both the -1 and

1 x values for the graph, which in turn only features onne clear intersection at point (0,0) on the graph instead.

## Basic Equation

In an inverse cubic equation, or any function with the cubed root of x, will be affected by the variables plugged into it (as its normal counterpart), which ultimately will determine how the graph looks. A number may change the direction of the graph, the slope of the curve, the turning point and the intercepts. For example, in the equation y = a(b(x – h))(1/3) + k, changing any of the variables, such as: a, b, h, or k will alter the graph in multiple ways, whether in length, amplitude, and even intersects.The basic equation, then, for any inverse cubic function is y = x(1/3).

For this very specific, basic example of the equation for an inverse function is only x. The values that can be plugged into the equation, being a cubed root and thus allowing answers ranging from being positive or negative, can be either negative or positive. When plugging in a positive X value into the equation, Y will always give you a ppositive number, and the same rule for negative numbers as well. Zero (0) is the only number here which follows another rule, and for which its cube root will always be zero. . This is why the shape of the graph takes the shape that it primarily has, the stretched out ¨s¨shape.

The turning point in the graph is also due to the equation. For example, if you plug in -1 in the equation then the value for y will be -1. If you plug in 0 for x then the y value will be 0. And, finally, if you plug in 1 for x then the value for y will give 1. This demonstrates the turning point in the graph, through points: (-1,-1), (0,0), and (1,1).

There exist currently no restrictions on the value of x that can go againts the inverse function, or that can give a result which is not measurable.

## Key Features

- No minimum or maximum values for y.- Horizontaly asymptotic.

## Domain, Range, Intercepts, and Vertices:

DomainRangeInterceptsy- intercept: (0,0)

## Real- World Applications

- Making Box-Height of Water in Spherical Tank-The Smallest Distance from a Parabola-Pumping Water out of Tank-Equation of State for Real Gases-Electrical Resistance-Finding Interest Rate-Break-Even Points in Economics-Von Neumann’s Model of an Expanding Economy-Trisecting the Angle- After the proof of Feramat’s Last Theorem ,Andrew Wiles, Open Problem for the Next Century

All of these are very clear real- world applications for the Cubic Function and its equation, although some are more complicated than others. IN essence, though, a cubic function would probably be used more often to find the volume or determine the volume quantity of multiple objects. This con not only help in determining those values superficially, but for ther related aspects such as bottle production (amount of volume to fit liquid inside) or in the creation of boxes, which are tools essential to the packaging and transportation of other materials. At simple sight a cubic function might not seem relevant at all, but to actually has a great dela to do with everyday measurements and tasks, and can help these in more ways than we could think of.

## Transformations

Let's now talk about how transformations affect a cubic function (shifts, streches and reflections). To understand how the Transformations affect the fucntion itself, we are going to work with the following example. The equation is y = a(b(x – h))^3 + k, so we are going to see how a, b, h and k affect the cubic functions graph.Let's set the cubic function as f(x) =

. The original Graph looks like this:1/1*(x – 0)^3 + 0We are going to first change the "k" value of the graph, which is outside the argument of the function. As we make this changes, the graph moves either up or down. Here is an example of the graph if we have a positive number for k:

As we can see, if we have h equal 5, then the whole graph Shifts 5 coordinates upward. The same pattern would happen if we subtract 5, but the grpah instead would move down 5 coordinates.

Now lets explore what happens if we change the "h" value. In this case, the number is within the argument. If we have a positive number for h, then that means we are subtracting within the argument. If we have a negative value for h, than we are adding within the arugment (because a number minus a negative is adding). So, if we add within the argument, the graph "Shifts" to the left. If we subtract within the argument, than the whole parabola moves to the right. Here is an example of this phenomena:

As we can appreciate, if we subtract 5 inside the argument, the whole parabola moves 5 coordinates to the right. Its kind of tricky to think of it this way, that adding means it moves to the left and subtracting means it moves to the right. But it makes sense because, if we want to add when we have a subtracting symbol we plug in a negative number. If we want to subtract when we have a subtracting symbol, then we plug in a positive number. Hence, if H is positive, the parabola moves to the left. If h is a negative, then the parabola moves to the right.

The next type of transformations we can deal with are "reflections" and "stretches". If we change the value of "a" of the function, the parabola tends to open or close tightly together. For instance, if we have a be equal a negative number, the smaller the negative number is the closer together it looks. Here is a graph of when "a" equals a negative number:

As we can see, the graph got closer together. Now, another important concept of changing "a" is that, if you change the sign of "a", the parabola will be reflected and the direction of the parabola would be opposite.

Finally, we have "b", which is a constant which is multiplied within the argument of the function. This constant also detemrines how strecthed the graph looks. The bigger, and positive the value for "b" is, the more the graph stretches. Here is an example:

As it can be appreciated, the parabola is more opened than the orginial one. This type of transformation is called a "stretch", because it causes the graph get wider, or thinner.

From this oobservations, we can conclude general rules for transformations. This rules are what causes the different changes in the graph of the cubic (and other) fucntions. The rules are:

f(x) +aisf(x) shifted upwardaunitsf(x) –aisf(x) shifted downwardaunitsf(x+a) isf(x) shifted leftaunitsf(x–a) isf(x) shifted rightaunitsf(x) isf(x) flipped upside down ("reflected about thex-axis")f(–x) is the mirror off(x) ("reflected about they-axis")f(ax) determines how wide the graph looksCitation1. Cubic Function Graph: http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/wynne/cubic/image3.gif

2. Inverse Cubic Function Graph: http://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/Images/cubeRoot.gif

3. Real- World Applications: http://archives.math.utk.edu/ICTCM/VOL10/C003/paper.pdf

4. Purple Math Transformations: http://www.purplemath.com/modules/fcntrans2.htm

5. Interactive Function Flyer; http://www.shodor.org/interactivate/activities/FunctionFlyer/

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