Graph The Function:${ R(x) = \left{ \begin{array}{ll} -x^2 + 4 & \text{for } X \leq 2 \ 2x - 4 & \text{for } X \ \textgreater \ 2 \end{array} \right. }$Part 1 Of 5:The First Rule Defines A Downward-opening Parabola With The

Introduction

In this article, we will delve into the world of piecewise functions and explore the graph of the given function $r(x) = \left\{ \begin{array}{ll} -x^2 + 4 & \text{for } x \leq 2 \\ 2x - 4 & \text{for } x \ \textgreater \ 2 \end{array} \right.$. This function is a classic example of a piecewise function, which is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function is typically defined using a set of rules, where each rule specifies the sub-function to be used for a particular interval of the domain. In the case of the given function, we have two sub-functions:

• For $x \leq 2$, the function is defined as $-x^2 + 4$.
• For $x > 2$, the function is defined as $2x - 4$.

Graphing the First Rule: $-x^2 + 4$

The first rule defines a downward-opening parabola with its vertex at $(0, 4)$. To graph this function, we can start by identifying the vertex and the direction of the parabola. Since the parabola opens downward, the function will decrease as we move away from the vertex.

Key Features of the First Rule

• Vertex: The vertex of the parabola is at $(0, 4)$.
• Direction: The parabola opens downward.
• Axis of Symmetry: The axis of symmetry is the vertical line $x = 0$.
• Intercepts: The parabola intersects the x-axis at $x = 2$ and $x = -2$.

Graphing the Second Rule: $2x - 4$

The second rule defines a linear function with a slope of 2 and a y-intercept of -4. To graph this function, we can start by identifying the y-intercept and the slope. Since the slope is positive, the function will increase as we move to the right.

Key Features of the Second Rule

• Y-intercept: The y-intercept of the line is at $y = -4$.
• Slope: The slope of the line is 2.
• Intercepts: The line intersects the x-axis at $x = 2$.

Graphing the Piecewise Function

To graph the piecewise function, we need to combine the graphs of the two rules. We can start by graphing the first rule, which is the downward-opening parabola. Then, we can graph the second rule, which is the linear function. Finally, we can combine the two graphs to get the final graph of the piecewise function.

Key Features of the Piecewise Function

• Domain: The domain of the function is all real numbers.
• Range: The range of the function is all real numbers.
• Intercepts: The function intersects the x-axis at $x = 2$ and $x = -2$.
• Asymptotes: The function has a vertical asymptote at $x = 2$.

Conclusion

Q&A: Graphing the Piecewise Function

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

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 the graphs to get the final graph of the piecewise function.

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

A: The key features of a piecewise function include its domain, range, intercepts, and asymptotes.

Q: How do I find the domain of a piecewise function?

A: To find the domain of a piecewise function, you need to identify the intervals of the domain for each sub-function and then combine the intervals to get the final domain of the function.

Q: How do I find the range of a piecewise function?

A: To find the range of a piecewise function, you need to identify the range of each sub-function and then combine the ranges to get the final range of the function.

Q: What are the intercepts of a piecewise function?

A: The intercepts of a piecewise function are the points where the function intersects the x-axis and y-axis.

Q: What are the asymptotes of a piecewise function?

A: The asymptotes of a piecewise function are the lines that the function approaches as x approaches positive or negative infinity.

Q: How do I graph a downward-opening parabola?

A: To graph a downward-opening parabola, you need to identify the vertex and the direction of the parabola. The vertex is the point where the parabola turns, and the direction is the direction in which the parabola opens.

Q: How do I graph a linear function?

A: To graph a linear function, you need to identify the y-intercept and the slope of the line. The y-intercept is the point where the line intersects the y-axis, and the slope is the rate at which the line rises or falls.

Q: How do I combine the graphs of two sub-functions?

A: To combine the graphs of two sub-functions, you need to graph each sub-function separately and then combine the graphs to get the final graph of the piecewise function.

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

A: Some common mistakes to avoid when graphing a piecewise function include:

• Graphing the wrong sub-function for a particular interval of the domain.
• Failing to identify the key features of the function, such as the domain, range, intercepts, and asymptotes.
• Not combining the graphs of the sub-functions correctly to get the final graph of the piecewise function.

Conclusion

In this article, we have answered some common questions about graphing piecewise functions. We have discussed the key features of a piecewise function, including its domain, range, intercepts, and asymptotes. We have also provided tips and tricks for graphing piecewise functions, including how to graph a downward-opening parabola and a linear function. Finally, we have identified some common mistakes to avoid when graphing a piecewise function.