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Chapter 3: Problem 46

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{-x} & {\text { if } x \leq 0} \\ {9-x^{2}} & {\text { if } 0< x \leq 3} \\ {x-3} & {\text { if } x>3}\end{array}\right.\)

Short Answer

Expert verified
Graph consists of a line, a parabola, and another line; each drawn in specified intervals.

Step by step solution

01

Analyze the First Piece

The first piece of the function is \(-x\) for \(x \leq 0\). This is a linear function with a slope of -1. It is defined for all values of \(x\) that are less than or equal to 0. When we sketch this piece, we start from the origin at \(x=0\) and move leftward along a line with a slope of -1. The graph is a straight line downward from the point \( (0,0) \).
02

Analyze the Second Piece

The second piece of the function is \(9-x^2\) for \(0 < x \leq 3\). This piece represents a downward-opening parabola. At \(x=0\), the value is excluded, creating a hollow point. Calculate a few values to sketch the curve: \(f(1) = 8, f(2) = 5, f(3) = 0\). The highest point of the curve is (0,9), but it's just a reference here as we only consider \(x > 0\). Draw a parabola opening downward from just above \((0,9)\) to the endpoints at (x,9-x^2) with a visible point at (x=3,0).
03

Analyze the Third Piece

The third piece is \(x-3\) for \(x>3\), another linear function with a slope of 1, which starts from (x=3) at (f(3) = 0). Sketch the line starting from the point (3,0) moving to the right with a slope of 1. This part of the graph will continue upward indefinitely as x increases.
04

Combine the Pieces

To sketch the complete graph of the piecewise function, combine all the pieces: Draw the line \(-x\) for \(x\leq0\), the downward parabola \(9-x^2\) for \(03\) starting from the point \((3,0)\), continuing upward. Ensure the transition between pieces is smooth and mark any open or closed points appropriately.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Piecewise Functions
Creating a graph of a piecewise defined function involves piecing together different functions. Each piece is restricted to a specific interval of x-values. By carefully plotting each segment, we can see how they connect to form the entire function.
  • Piecewise functions often consist of various linear and non-linear equations, each with a designated domain.
  • It's crucial to pay attention to open and closed dots at the boundaries of each segment. This indicates whether a boundary value is included or not.
  • The challenge lies in ensuring smooth transitions between pieces without any overlap or gaps.
When graphing piecewise functions, first label the endpoints and check the continuity of the graph across these points. This will help in ensuring you correctly represent the function鈥檚 structure.
Combining Function Pieces
To visualize a piecewise function, you must correctly combine its segments. Each piece operates independently over its specified range. They join like "puzzle pieces" to form a complete graph.
  • Start by reviewing the intervals provided for each piece. This helps in understanding where each function is active in the graph.
  • Plot each segment individually, using values within its range, to determine the shape and direction.
  • Make sure to plot open or closed dots based on whether the interval includes the endpoint (such as using a filled dot for 鈮� or 鈮� and an open dot for < or >).
By clearly marking transitions between pieces, the combined graph becomes a smooth and accurate representation of the function. Checking for any misalignment at the boundaries is also key to ensuring accuracy.
Linear Functions in Pieces
Linear functions in a piecewise graph are simple lines that you can easily plot once the slope and y-intercept are known. Each part operates over a specific interval and should be differentiated by its direction and boundary conditions.
  • Identify linear pieces in the function. They are usually in the form of \(y = mx + b\).
  • The slope \(m\) tells the line's steepness and direction. For example, \(-x\) indicates a line going downward, while \(x-3\) slopes upwards.
  • Each linear part begins or ends at specified x-values. Pay attention to whether these points are included (closed dots) or not (open dots).
By applying these rules, linear functions within piecewise graphs become straightforward to sketch.
Parabolic Curves in Pieces
Parabolic curves add complexity to piecewise functions due to their curved, non-linear nature. They often represent sections of a quadratic graph within a specific interval.
  • Usually represented by an equation like \(y = ax^2 + bx + c\), in piecewise functions, they have notable endpoints dependent on their interval.
  • These functions may not include endpoints, as shown with open dots, such as the top of our parabola, \(9-x^2\), beginning just above (0,9).
  • Sketching involves calculating key points on the curve. Stay aware of how they bend and the area they cover on the graph.
Understanding the unique properties of parabolic sections will allow you to seamlessly integrate them into your overall piecewise graph.

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