Problem 31 - Applied Calculus

Question

31-32. Use a graphing calculator to graph each piecewise nonlinear function on the window [-2,10] by [-5,5] . Where parts of the graph do not touch, state which point is included and which is excluded. $$ f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \leq 2 \\ 6-x & \text { if } 2 < x < 6 \\ 2 x-17 & \text { if } x \geq 6 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the pieces separately, (2,4) included; (2,4) excluded for line, (6,-5) included.

Step 1: Understanding Piecewise Functions

We have a piecewise function defined with three rules based on the value of \(x\). For \(x \leq 2\), \(f(x) = x^2\). For \(2 < x < 6\), \(f(x) = 6 - x\). For \(x \geq 6\), \(f(x) = 2x - 17\). Each function rule will create a different section of the graph depending on these intervals.

Step 2: Graph the First Piece

The first part \(f(x) = x^2\) applies when \(x \leq 2\). This is a parabolic curve opening upwards with a vertex at \( (0,0) \). On the interval from \(-2\) to \(2\), the parabola starts from \((4,16)\) at x=-2, moves through \((0,0)\), and ends at \((2, 4)\). The point \((2,4)\) is included, marked by a solid dot on the graph.

Step 3: Graph the Second Piece

For \(2 < x < 6\), \(f(x) = 6 - x\). This is a linear function with a slope of -1, starting just beyond \(x=2\). At \(x = 2\), the value is \(4\), and at \(x = 6\), it reaches 0. Thus it goes through \((2, 4)\) with an open point (not included) and \((6, 0)\) also as an open point.

Step 4: Graph the Third Piece

For \(x \geq 6\), \(f(x) = 2x - 17\). This function is linear with a slope of 2, beginning at \((6, -5)\) and continuing. The point \((6, -5)\) is included, shown by a solid dot, and the line extends rightward from there.

Step 5: Identifying Included and Excluded Points

Review the transitions at the boundaries of each piece: At \(x=2\), the point on the parabola \((2,4)\) is included, while on the line \((-x + 6)\), it is considered not included. At \(x=6\), the \(6-x\) line value \((6,0)\) is not included, while \((6,-5)\) on the \(2x-17\) line is included.

Key Concepts

Understanding and Graphing Piecewise Functions

When working with piecewise functions, you will notice that these are a set of multiple functions, each given a specific interval over which it is applied. In our piecewise function example, we have three separate functions, each with its own domain:

• For \(x \leq 2\), the function is \(f(x) = x^2\), a quadratic curve.
• For \(2 < x < 6\), we use \(f(x) = 6 - x\), a linear function with a negative slope.
• Finally, for \(x \geq 6\), we have another linear function, \(f(x) = 2x - 17\), with a positive slope.

To graph piecewise functions effectively:

• Draw each segment of the function separately, observing the specified intervals.
• Mark endpoints carefully to show inclusion (solid dots) or exclusion (open dots).
• Check each transition between segments for continuity and the inclusion or exclusion of boundary values.

Each function creates a distinct portion of the graph based on the conditions of the interval, reflecting changes in slope and curvature.

Exploring Piecewise Nonlinear Functions

Piecewise nonlinear functions can be more complex to graph due to the different types of equations involved. In our example, \(f(x) = x^2\) is a nonlinear function within the interval \(x \leq 2\). This type of function forms part of our piecewise function's graph as a parabola.Working with these functions requires:

• Identifying the vertex and shape of the nonlinear graph segment, like the parabola opening upwards from \(x \leq 2\) that travels from \((-2, 4)\) to \((2,4)\).
• Ensuring smooth curves and avoiding abrupt jumps within the individual sections, unless transitioning between different function rules.

Remember, nonlinear segments can significantly change the style of the graph, and understanding each part is fundamental to sketch the entire piecewise function accurately.

Interpreting Graphs of Piecewise Functions

Understanding a piecewise function graph involves reading and interpreting each segment's characteristics. Let's break down how to interpret the segments and transitions effectively:

• Acknowledge how each sub-function behaves within its interval. The initial quadratic segment indicates a gradual change, portraying an ascending curve until \(x = 2\).
• Notice the linear portion \(6 - x\), reflecting a steady decline from \(x = 3\) to just before \(x = 6\). This part illustrates how a constant linear decrease looks over its interval. Both endpoints \((2, 4)\) and \((6, 0)\) are represented as open circles in the graph.
• The final equation \(2x - 17\) begins from \(x = 6\) with a significant increase, forming a straight line with a steep slope. The transition at \(x = 6\) involves a shift from a closed to open point, clearly representing the inclusion/exclusion of certain values.

Smoothly transitioning between the function parts and understanding included versus excluded points helps make sense of the graph as a cohesive representation of the piecewise function.