Problem 69 - Thomas Calculus

Question

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(0)=0, f(1)=2, f(-1)=-2, \lim _{x \rightarrow-\infty} f(x)=-1,\text{ and }\\\\{\lim _{x \rightarrow \infty} f(x)=1}$$

Step-by-Step Solution

Answer

Sketch with points at (0,0), (1,2), (-1,-2), asymptotes at y=-1, y=1.

Step 1: Plot Given Points

Start by plotting the specific points provided in the conditions on a coordinate plane: 1. The point \((0, 0)\) is the origin,.2. The point \((1, 2)\) lies to the right of the origin, above the x-axis.3. The point \((-1, -2)\) lies to the left of the origin, below the x-axis.

Step 2: Examine the Limits

Interpret the given limits for behavior as \(x\) approaches infinity:1. \(\lim _{x \rightarrow -\infty} f(x) = -1\) indicates that as \(x\) becomes very negative, \(f(x)\) should approach \(y = -1\). Therefore, draw a horizontal asymptote at \(y = -1\) as \(x\) decreases.2. \(\lim _{x \rightarrow \infty} f(x) = 1\) indicates that as \(x\) becomes very positive, \(f(x)\) should approach \(y = 1\). Therefore, draw a horizontal asymptote at \(y = 1\) as \(x\) increases.

Step 3: Sketch a Possible Graph

Sketch a function that goes through the plotted points and corresponds with the asymptotic behavior obtained from the limits.1. Start the curve from the left, trending toward the horizontal asymptote at \(y = -1\) approaching the point \((-1, -2)\).2. Continue through the origin \((0, 0)\) steadily increasing to meet the point \((1, 2)\).3. Extend the graph to the right, rising smoothly towards the horizontal asymptote at \(y = 1\) as \(x\) continues to increase.

Step 4: Verify the Graph

Check that the graph passes through all the specified points and that it respects the behavior described by the limit conditions. Ensure the transitions between plotted points and asymptotes appear smooth and continuous without abrupt changes in direction.

Key Concepts

Asymptotic Behavior

Asymptotic behavior helps us understand how a function behaves as its input values approach infinity or negative infinity. It's like peeking into the future of your graph at the extreme ends.
A function's graph approaches a line, known as an asymptote, but it never actually touches it.
In this exercise, we're given two horizontal asymptotes:

• As \(x\) approaches negative infinity, \(f(x)\) moves toward the line \(y = -1\).
• As \(x\) approaches positive infinity, \(f(x)\) moves toward the line \(y = 1\).

When sketching a graph, imagine that the curve gets closer and closer to these asymptotes but never really meets them.
This behavior tells us a lot about where the function is heading, even if it may change drastically between these trends.

Coordinate Axes

The coordinate axes are the horizontal and vertical lines on a graph that form the foundation for plotting functions and points. The horizontal line is called the x-axis, and the vertical line is the y-axis.
Every function's graph is plotted with respect to these axes. Think of them as the skeleton of your graph, giving it direction and placement.
To plot points like \((0, 0)\), \((1, 2)\), and \((-1, -2)\), you begin at the origin where the x-axis and y-axis intersect (this is the point \( (0,0) \)).

• \((0, 0)\) lies at the very intersection.
• \((1, 2)\) is one unit right and two units up.
• \((-1, -2)\) is one unit left and two units down.

These points act as guideposts for your sketch, ensuring that your function graph is correctly situated on the plane.

Limit of a Function

The limit of a function tells us what value a function approaches as the input approaches a particular point or infinity. Limits are a cornerstone of calculus and help us understand the behavior of functions at borders.
In our exercise, we examine the limits as \(x\) approaches extreme values:

• If \(x\) approaches negative infinity, \(f(x)\) tends to \(y = -1\).
• If \(x\) approaches positive infinity, \(f(x)\) tends to \(y = 1\).

These limits create horizontal asymptotes, which are essential for sketching the overall shape of the graph. Understanding limits plays a vital role in predicting and drawing where a graph is headed without plotting every single point.

Continuous Functions

Continuous functions are those that have no breaks, gaps, or jumps in their graphs. These functions flow smoothly from point to point, creating an unbroken line or curve.
In this exercise, when sketching the graph, we aim for a continuous transition between the plotted points and the asymptotes. This means our graph should look seamless as it moves from point \((-1, -2)\) through \((0, 0)\) to \((1, 2)\), and onward.
Ensuring continuity also means that there are no abrupt or sharp changes in direction. Instead, the graph should gently curve according to the behavior dictated by the points and asymptotes.

• A smooth transition from one behavior to another is vital to maintain the overall integrity and correctness of the graph.

By understanding continuity, you'll better appreciate how different parts of a function relate and interact across a graph.