Complete The Table To Vertically Stretch The Parent Function $f(x)=x^3$ By A Factor Of 3.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y = K F(x) = 3x^3$ \\ \hline 0 & 0 \\ 1 & $\square$ \\ 2 & $\square$

Introduction

In mathematics, functions are used to describe the relationship between variables. When we talk about stretching a function, we are referring to the process of modifying the function's graph by scaling it vertically or horizontally. In this article, we will focus on vertically stretching the parent function $f(x)=x^3$ by a factor of 3. This involves creating a new function that is a multiple of the original function, which will result in a new graph that is a scaled version of the original graph.

Understanding the Parent Function

Before we can stretch the parent function, we need to understand what it looks like. The parent function $f(x)=x^3$ is a cubic function that has a graph that opens upwards. It has a single turning point, which is the vertex of the graph. The vertex of the graph is located at the origin (0,0), and the graph has a positive slope.

Vertically Stretching the Parent Function

To vertically stretch the parent function $f(x)=x^3$ by a factor of 3, we need to multiply the function by 3. This will result in a new function that is a multiple of the original function. The new function is given by:

$$y = k f(x) = 3x^3$$

where $k$ is the scaling factor. In this case, $k=3$, so the new function is:

$$y = 3x^3$$

Creating the Table

To create the table, we need to plug in the values of $x$ into the new function and calculate the corresponding values of $y$. The table will have two columns: one for the values of $x$ and one for the values of $y$. The table will look like this:

Filling in the Table

To fill in the table, we need to plug in the values of $x$ into the new function and calculate the corresponding values of $y$. For $x=0$, we have:

$$y = 3(0)^3 = 0$$

So, the first row of the table is complete. For $x=1$, we have:

$$y = 3(1)^3 = 3$$

So, the second row of the table is complete. For $x=2$, we have:

$$y = 3(2)^3 = 24$$

So, the third row of the table is complete.

The Completed Table

The completed table looks like this:

Conclusion

In this article, we have vertically stretched the parent function $f(x)=x^3$ by a factor of 3. We have created a new function that is a multiple of the original function, and we have filled in the table with the corresponding values of $y$. The completed table shows that the new function has a graph that is a scaled version of the original graph.

Discussion

Vertically stretching a function is a common technique used in mathematics to create new functions that are related to the original function. It is an important concept in algebra and calculus, and it has many applications in science and engineering. In this article, we have focused on vertically stretching the parent function $f(x)=x^3$ by a factor of 3, but the technique can be applied to any function.

Examples

Here are some examples of vertically stretching functions:

• Vertically stretching the parent function $f(x)=x^2$ by a factor of 2 results in the function $y = 2x^2$.
• Vertically stretching the parent function $f(x)=x^4$ by a factor of 3 results in the function $y = 3x^4$.
• Vertically stretching the parent function $f(x)=\sin x$ by a factor of 2 results in the function $y = 2\sin x$.

Applications

Vertically stretching functions has many applications in science and engineering. For example:

• In physics, vertically stretching a function can be used to model the motion of an object under the influence of gravity.
• In engineering, vertically stretching a function can be used to design systems that are scalable and efficient.
• In economics, vertically stretching a function can be used to model the behavior of economic systems.

Conclusion

In conclusion, vertically stretching the parent function $f(x)=x^3$ by a factor of 3 is a simple yet powerful technique that can be used to create new functions that are related to the original function. The technique has many applications in science and engineering, and it is an important concept in algebra and calculus.