Graph The Following Piecewise Function. F ( X ) = { X 2 + 2 If − 5 ≤ X \textless 3 X − 4 If 3 ≤ X \textless 7 F(x) = \begin{cases} x^2 + 2 & \text{if } -5 \leq X \ \textless \ 3 \\ x - 4 & \text{if } 3 \leq X \ \textless \ 7 \end{cases} F ( X ) = { X 2 + 2 X − 4 ​ If − 5 ≤ X \textless 3 If 3 ≤ X \textless 7 ​

Introduction

Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will focus on graphing a piecewise function, specifically the function $f(x) = \begin{cases} x^2 + 2 & \text{if } -5 \leq x \ \textless \ 3 \\ x - 4 & \text{if } 3 \leq x \ \textless \ 7 \end{cases}$.

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each of which is defined on a specific interval. The intervals are usually separated by a set of points, called the "break points" or "transition points". The function is then defined as a combination of these sub-functions, with each sub-function being applied to the corresponding interval.

Graphing Piecewise Functions

To graph a piecewise function, we need to graph each sub-function separately and then combine them. Here's a step-by-step guide on how to graph the given piecewise function:

Graphing the First Sub-Function

The first sub-function is $f(x) = x^2 + 2$, which is defined on the interval $-5 \leq x \ \textless \ 3$. To graph this function, we need to find the x-intercepts, the y-intercept, and the vertex of the parabola.

• X-Intercepts: To find the x-intercepts, we need to set $f(x) = 0$ and solve for x. This gives us the equation $x^2 + 2 = 0$, which has no real solutions. Therefore, the x-intercepts are not defined.
• Y-Intercept: To find the y-intercept, we need to evaluate $f(0)$. This gives us $f(0) = 0^2 + 2 = 2$. Therefore, the y-intercept is (0, 2).
• Vertex: To find the vertex, we need to use the formula $x = -\frac{b}{2a}$. In this case, $a = 1$ and $b = 0$, so the vertex is at $x = -\frac{0}{2(1)} = 0$. Evaluating the function at this point, we get $f(0) = 0^2 + 2 = 2$. Therefore, the vertex is (0, 2).

Graphing the Second Sub-Function

The second sub-function is $f(x) = x - 4$, which is defined on the interval $3 \leq x \ \textless \ 7$. To graph this function, we need to find the x-intercepts, the y-intercept, and the slope of the line.

• X-Intercepts: To find the x-intercepts, we need to set $f(x) = 0$ and solve for x. This gives us the equation $x - 4 = 0$, which has a solution of x = 4. Therefore, the x-intercept is (4, 0).
• Y-Intercept: To find the y-intercept, we need to evaluate $f(0)$. This gives us $f(0) = 0 - 4 = -4$. Therefore, the y-intercept is (0, -4).
• Slope: To find the slope, we need to use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. We can choose any two points on the line, such as (3, -1) and (7, 3). Evaluating the formula, we get $m = \frac{3 - (-1)}{7 - 3} = \frac{4}{4} = 1$. Therefore, the slope is 1.

Combining the Sub-Functions

Now that we have graphed each sub-function separately, we can combine them to get the final graph of the piecewise function. The graph consists of two parts: a parabola defined on the interval $-5 \leq x \ \textless \ 3$ and a line defined on the interval $3 \leq x \ \textless \ 7$.

Conclusion

Graphing piecewise functions can be a challenging task, but by breaking it down into smaller sub-functions and graphing each one separately, we can make the process more manageable. In this article, we graphed the piecewise function $f(x) = \begin{cases} x^2 + 2 & \text{if } -5 \leq x \ \textless \ 3 \\ x - 4 & \text{if } 3 \leq x \ \textless \ 7 \end{cases}$ and obtained the final graph.

Key Takeaways

• Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval.
• To graph a piecewise function, we need to graph each sub-function separately and then combine them.
• The graph of a piecewise function consists of multiple parts, each corresponding to a different sub-function.
• By breaking down the graphing process into smaller sub-functions, we can make the process more manageable.

Further Reading

For more information on graphing piecewise functions, we recommend checking out the following resources:

• Khan Academy: Piecewise Functions
• Math Is Fun: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

References

• [1] Larson, R. (2013). Calculus. Brooks Cole.
• [2] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2016). Calculus: A New Horizon. John Wiley & Sons.

Introduction

Graphing piecewise functions can be a challenging task, but with the right guidance, it can be made easier. In this article, we will provide a Q&A guide to help you understand and graph piecewise functions.

Q: What is a piecewise function?

A: A piecewise function is a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval. These sub-functions are usually separated by a set of points, called the "break points" or "transition points".

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function separately and then combine them. Here's a step-by-step guide:

1. Identify the sub-functions and their corresponding intervals.
2. Graph each sub-function separately, using the standard graphing techniques for each type of function (e.g. parabolas, lines, etc.).
3. Combine the sub-functions by drawing a single graph that includes all the sub-functions.

Q: What are the key characteristics of a piecewise function?

A: The key characteristics of a piecewise function are:

• Multiple sub-functions, each defined on a specific interval.
• Break points or transition points that separate the sub-functions.
• A single graph that includes all the sub-functions.

Q: How do I determine the break points of a piecewise function?

A: To determine the break points of a piecewise function, you need to identify the points where the sub-functions change. These points are usually specified in the function definition.

Q: Can I use technology to graph piecewise functions?

A: Yes, you can use technology to graph piecewise functions. Many graphing calculators and computer software programs, such as Desmos and GeoGebra, can graph piecewise functions.

Q: What are some common mistakes to avoid when graphing piecewise functions?

A: Some common mistakes to avoid when graphing piecewise functions include:

• Failing to identify the sub-functions and their corresponding intervals.
• Graphing the sub-functions incorrectly.
• Failing to combine the sub-functions correctly.

Q: How do I check my work when graphing piecewise functions?

A: To check your work when graphing piecewise functions, you can:

• Verify that the sub-functions are correct.
• Check that the break points are correct.
• Make sure that the graph includes all the sub-functions.

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

A: Piecewise functions have many real-world applications, including:

• Modeling population growth and decline.
• Analyzing financial data.
• Studying the behavior of physical systems.

Conclusion

Graphing piecewise functions can be a challenging task, but with the right guidance, it can be made easier. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in graphing piecewise functions.

Key Takeaways

• Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval.
• To graph a piecewise function, you need to graph each sub-function separately and then combine them.
• The key characteristics of a piecewise function are multiple sub-functions, break points, and a single graph that includes all the sub-functions.

Further Reading

For more information on graphing piecewise functions, we recommend checking out the following resources:

• Khan Academy: Piecewise Functions
• Math Is Fun: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

References

• [1] Larson, R. (2013). Calculus. Brooks Cole.
• [2] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2016). Calculus: A New Horizon. John Wiley & Sons.

Note: The references provided are a selection of popular calculus textbooks that cover piecewise functions. They are not exhaustive and are intended to provide additional resources for further reading.