Question

Sketching graphs Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\begin{aligned}&f(-1)=-2, f(1)=2, f(0)=0, \lim _{x \rightarrow \infty} f(x)=1\\\&\lim _{x \rightarrow-\infty} f(x)=-1 \end{aligned}$$

Short answer

Question: Sketch a graph for a function f that satisfies the following conditions: passing through points (-1, -2), (1, 2), and (0, 0), having a limit of 1 as x approaches infinity, and having a limit of -1 as x approaches negative infinity. Answer: To sketch the graph for the given function, we need to perform the following steps: 1. Plot the points (-1, -2), (1, 2), and (0, 0) on the graph. 2. Determine the horizontal asymptotes based on given limits: y = 1 as x approaches infinity and y = -1 as x approaches negative infinity. 3. Connect the points while considering the asymptotes and pass through each given point - moving upwards to the right and downwards to the left from (-1,-2), maintaining the function's continuity. 4. Finalize the graph by ensuring smooth connections between points and asymptotes, satisfying all the conditions provided.

Step-by-step solution

Plot the given points

Plot the points \((-1, -2),\) \((1, 2),\) and \((0, 0)\) on the graph.

Determine horizontal asymptotes

From the exercise, we know that as x approaches infinity, f(x) approaches 1 and as x approaches negative infinity, f(x) approaches -1. These will be our horizontal asymptotes: $$y = 1$$ and $$y = -1$$

Connect the points

Start at \((-1, -2)\) and move upwards to the right, passing through \((0, 0)\) and reaching the horizontal asymptote \(y=1\). Go downwards as we move to the left, passing through \((0,0)\) again, and move towards the horizontal asymptote \(y=-1\).

Finalize the graph

Draw the connections between points and asymptotes smoothly while maintaining the function's continuity. The final graph should satisfy all the conditions and incorporate the given points and horizontal asymptotes.

Key concepts

Horizontal Asymptotes

Understanding horizontal asymptotes is crucial when graphing functions, especially in calculus. A horizontal asymptote is a horizontal line that the graph of a function approaches as the input, or x-value, goes to infinity or negative infinity. It represents the value that the function will get infinitely close to but generally never reach.

In the given exercise, you are told that the function approaches 1 as x approaches positive infinity, and it approaches -1 as x approaches negative infinity. This tells us that the horizontal asymptotes for our function are the lines \( y = 1 \) and \( y = -1 \). These asymptotes act as a boundary that the function will never cross as x becomes very large or very small.

When sketching graphs, you should always start by identifying and marking the horizontal asymptotes because they provide a structural backbone for graph behavior at the extremities of the x-axis.

Graphing Functions

Graphing functions is a visual way to understand the behavior of mathematical functions. It's a core concept in calculus and other areas of mathematics. The key steps include plotting points, identifying important features like intercepts and asymptotes, and noting the smoothness and continuity of the graph.

As per the given conditions in our exercise, three specific points are plotted: \((-1, -2)\), \((1, 2)\), and \((0, 0)\). These points give us a skeleton around which to build our graph. The limits at infinity, defined by the horizontal asymptotes, guide us on how the function behaves as it stretches towards infinity. When you connect these points, ensure to do so with a curve that respects the asymptotic behavior: it must get close to but not touch or cross the horizontal asymptotes.

Exercise Improvement Advice

When drawing the curve, imagine the function's smooth path, transitioning through the plotted points and stretching toward the horizontal asymptotes with curvature that matches the nature of the function. Always recheck that every condition is met—that your function goes through the given points and aligns with the predicted limits at infinity.

Limits at Infinity

Limits at infinity refer to the value that a function approaches as the input increases or decreases without bound. This concept is closely linked with horizontal asymptotes. When calculating the limit of a function as x goes to infinity, you are essentially trying to figure out where the function 'levels off' at extreme values of x.

In our exercise, the limits given suggest a specific behavior for the function at both ends of the x-axis. As x goes to positive infinity (to the right), the function approaches the value 1, and as x goes to negative infinity (to the left), it approaches the value -1. This information tells us that no matter how much we increase or decrease x, the y-values will get closer and closer to these two horizontal lines without actually reaching them. Showing this in a graph involves plotting the function so it snugs up against these asymptotes as x grows large in either direction.

When sketching graphs involving limits at infinity, always be mindful of how the function behaves as it moves away from the center (where x=0) and extends towards the edges of the graph. The smooth, approaching nature of the graph near the asymptotes is the graphical representation of the limits at infinity.