Problem 32 - Brief Applied Calculus

Question

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. $$ f(x)=\left\\{\begin{array}{ll} 4-x^{2} & \text { if } x<3 \\ 2 x-11 & \text { if } 3 \leq x<7 \\ 8-x & \text { if } x \geq 7 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph includes \((3, -5)\) (included), \((7, 3)\) (excluded), and \((7, 1)\) (included).

Step 1: Understand the Piecewise Function

The function is given in three parts based on the value of \(x\):- \(f(x) = 4 - x^2\) for \(x < 3\)- \(f(x) = 2x - 11\) for \(3 \leq x < 7\)- \(f(x) = 8 - x\) for \(x \geq 7\). Each expression describes the function within a specific range of \(x\) values.

Step 2: Graph the First Piece

For \(x < 3\), graph \(f(x) = 4 - x^2\). As this is a quadratic function, it will form a parabola opening downwards. The point (3, -5) should be noted as open (excluded) since \(x\) does not include 3.

Step 3: Graph the Second Piece

For \(3 \leq x < 7\), graph \(f(x) = 2x - 11\). This is a linear equation, resulting in a straight line starting from the point (3, -5), which is included as it satisfies the inequality \(3 \leq x\). At \(x = 7\), the point (7, 3) is open since \(x\) is less than 7.

Step 4: Graph the Third Piece

For \(x \geq 7\), graph \(f(x) = 8 - x\). This is a linear equation, resulting in a straight line starting from (7, 1), which is included since \(x\) is greater than or equal to 7. Extend this line as \(x\) increases.

Step 5: Combine the Graphs

Combine these graphs on the window of \([-2, 10]\) by \([-5, 5]\). Ensure to mark open circles where points are excluded and closed circles where points are included, specifically (3, -5) as closed, (7, 3) and (7, 1) as open and closed respectively. Highlight transitions where the equation changes.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can be used to visualize mathematical functions. It goes beyond simple arithmetic and allows you to graph complex equations like piecewise functions on a given range. In this exercise, using a graphing calculator helps us plot the given piecewise function on a specified rectangular viewing window, from \([-2, 10]\) along the x-axis and \([-5, 5]\) on the y-axis.

To graph the function:

• Enter each piece of the function separately into the calculator's graphing feature.
• Make sure you carefully set the starting and ending points for each piece to match the specified inequality restrictions in the function. Such as entering for the range whether the point is included with 'closed' settings or an 'open' one.
• The graphing tool will help visualize the transitions and connections between the different sections of the piecewise function.

Using this tool can simplify understanding and ensure accuracy when plotting functions, as it visually represents the function's behavior over the specified range.

Quadratic Functions

Quadratic functions form the first part of many piecewise functions, as seen in this exercise. A quadratic function typically appears in the form of \( ax^2 + bx + c \), which forms a parabola when graphed. In the given exercise, for \( x < 3 \), the function segment \( f(x) = 4 - x^2 \) forms a downward-opening parabola starting from the values approaching \(x = 3\).

Characteristics of quadratic functions include:

• Vertex: The turning point of the parabola, which is located at \((0, 4)\) for the expression \(4 - x^2\).
• Axis of symmetry: A vertical line that divides the parabola into two symmetrical halves, with our example symmetrical about \(x = 0\).
• Direction: The opening direction is determined by the sign of \(a\). A negative \(a\), such as in \(-x^2\), opens downwards.

Understanding these features helps in sketching and analyzing the quadratic section of a piecewise function effectively.

Linear Functions

Linear functions are a key component in piecewise functions and are expressed in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In this task, two segments of the piecewise function are linear. First, \(3 \leq x < 7\) with the equation \(f(x) = 2x - 11\), and second, \(x \geq 7\) with the equation \(f(x) = 8 - x\).

For linear functions, note the following characteristics:

• Slope: Represents the steepness and direction of the line. For \(2x - 11\), the slope is 2, indicating an upward slant, while for \(8 - x\), the slope is -1, slanting downward.
• Y-intercept: The point where the line crosses the y-axis. The line \(f(x) = 2x - 11\) crosses at \((0, -11)\), and \(f(x) = 8 - x\) crosses at \((0, 8)\).
• Endpoints: Critical to plot correctly as per the conditions, with inclusion or exclusion as dictated by the inequality.

Each line segment contributes to the overall shape of the piecewise function, showing different linear behavior across its domain.