Describe How The Graph Of The Following Function Can Be Obtained From One Of The Basic Graphs:${ G(x) = -\frac{1}{3}(x-3)^2 }$1. Start With The Graph Of { Y = X^2 $}$.2. Shift It Horizontally By 3 Units To The Right.3. Vertically

Introduction

In mathematics, graphing functions is an essential skill that helps us visualize and understand the behavior of functions. One way to graph a function is by transforming a basic graph. In this article, we will describe how to obtain the graph of the function $g(x) = -\frac{1}{3}(x-3)^2$ from the basic graph of $y = x^2$.

Step 1: Start with the Graph of $y = x^2$

The graph of $y = x^2$ is a parabola that opens upwards. It has a minimum point at $(0, 0)$ and is symmetric about the y-axis. To start with this graph, we need to understand its shape and position.

Step 2: Shift the Graph Horizontally by 3 Units to the Right

To shift the graph of $y = x^2$ horizontally by 3 units to the right, we need to replace $x$ with $(x-3)$. This means that for every point $(x, y)$ on the original graph, the corresponding point on the new graph will be $(x-3, y)$. This shift will move the graph to the right by 3 units.

Step 3: Vertically Stretch the Graph

The function $g(x) = -\frac{1}{3}(x-3)^2$ has a coefficient of $-\frac{1}{3}$ in front of the squared term. This means that the graph will be vertically stretched by a factor of $-\frac{1}{3}$. To achieve this, we need to multiply the y-coordinate of each point on the graph by $-\frac{1}{3}$.

Combining the Transformations

Now that we have described the individual transformations, let's combine them to obtain the final graph. First, we shift the graph of $y = x^2$ horizontally by 3 units to the right. Then, we vertically stretch the graph by a factor of $-\frac{1}{3}$. The resulting graph will be the graph of $g(x) = -\frac{1}{3}(x-3)^2$.

Visualizing the Graph

To visualize the graph of $g(x) = -\frac{1}{3}(x-3)^2$, we can use a graphing calculator or software. The graph will be a parabola that opens downwards, with a maximum point at $(3, 0)$. The graph will be symmetric about the vertical line $x = 3$.

Conclusion

In this article, we described how to obtain the graph of the function $g(x) = -\frac{1}{3}(x-3)^2$ from the basic graph of $y = x^2$. We applied three transformations: shifting the graph horizontally by 3 units to the right, and vertically stretching the graph by a factor of $-\frac{1}{3}$. The resulting graph is a parabola that opens downwards, with a maximum point at $(3, 0)$. This example illustrates the importance of understanding the properties of basic graphs and how to transform them to obtain new graphs.

Key Takeaways

• To graph a function, we can start with a basic graph and apply transformations.
• Shifting a graph horizontally by $c$ units to the right involves replacing $x$ with $(x-c)$.
• Vertically stretching a graph by a factor of $k$ involves multiplying the y-coordinate of each point by $k$.
• Combining multiple transformations can help us obtain the graph of a complex function.

Further Exploration

• Experiment with different basic graphs and transformations to see how they can be combined to obtain new graphs.
• Use graphing software or calculators to visualize the graphs of different functions.
• Practice applying transformations to different functions to develop your skills in graphing.
  Frequently Asked Questions (FAQs) about Graphing Functions =============================================================

Q: What is the basic graph of $y = x^2$?

A: The basic graph of $y = x^2$ is a parabola that opens upwards. It has a minimum point at $(0, 0)$ and is symmetric about the y-axis.

Q: How do I shift the graph of $y = x^2$ horizontally by 3 units to the right?

A: To shift the graph of $y = x^2$ horizontally by 3 units to the right, you need to replace $x$ with $(x-3)$. This means that for every point $(x, y)$ on the original graph, the corresponding point on the new graph will be $(x-3, y)$.

Q: What is the effect of vertically stretching the graph of $y = x^2$ by a factor of $-\frac{1}{3}$?

A: Vertically stretching the graph of $y = x^2$ by a factor of $-\frac{1}{3}$ involves multiplying the y-coordinate of each point on the graph by $-\frac{1}{3}$. This will result in a graph that is vertically compressed by a factor of $-\frac{1}{3}$.

Q: How do I combine multiple transformations to obtain the graph of a complex function?

A: To combine multiple transformations, you need to apply each transformation in the correct order. For example, if you want to shift the graph of $y = x^2$ horizontally by 3 units to the right and then vertically stretch it by a factor of $-\frac{1}{3}$, you need to first replace $x$ with $(x-3)$ and then multiply the y-coordinate of each point by $-\frac{1}{3}$.

Q: What is the resulting graph of $g(x) = -\frac{1}{3}(x-3)^2$?

A: The resulting graph of $g(x) = -\frac{1}{3}(x-3)^2$ is a parabola that opens downwards, with a maximum point at $(3, 0)$. The graph is symmetric about the vertical line $x = 3$.

Q: How can I visualize the graph of a function?

A: You can visualize the graph of a function using a graphing calculator or software. These tools allow you to enter the function and see the resulting graph.

Q: What are some common transformations that can be applied to a graph?

A: Some common transformations that can be applied to a graph include:

• Shifting the graph horizontally by $c$ units to the right or left
• Shifting the graph vertically by $k$ units up or down
• Rotating the graph by $90^\circ$ clockwise or counterclockwise
• Reflecting the graph across the x-axis or y-axis
• Vertically stretching or compressing the graph by a factor of $k$

Q: How can I practice graphing functions?

A: You can practice graphing functions by:

• Using graphing software or calculators to visualize the graphs of different functions
• Creating your own functions and graphing them
• Working with a partner or tutor to practice graphing functions
• Using online resources and tutorials to learn more about graphing functions

Q: What are some real-world applications of graphing functions?

A: Graphing functions has many real-world applications, including:

• Modeling population growth or decline
• Analyzing the behavior of physical systems, such as springs or pendulums
• Optimizing business processes or supply chains
• Predicting the behavior of complex systems, such as weather patterns or financial markets

Conclusion

Graphing functions is an essential skill that has many real-world applications. By understanding the properties of basic graphs and how to transform them, you can create complex graphs that model real-world phenomena. Whether you're working in science, engineering, or business, graphing functions is an important tool that can help you analyze and understand complex systems.