Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(y=1-3 x^{-2}\)

Short answer

The function \(y = 1 - 3 x^{-2}\) has x-intercepts at \(-\sqrt{3}\) and \(\sqrt{3}\), a horizontal asymptote at \(y = 1\), a vertical asymptote at \(x=0\), and no extrema. The graph declines from \(y = 1\), intersects the x-axis at the x-intercepts, and then ascends back towards \(y = 1\), but it never crosses the y value of 1.

Step-by-step solution

Find the intercepts

To find the x-intercept, set \(y = 0\) and solve for \(x\). To find the y-intercept, set \(x = 0\) and solve for \(y\). However, note that \(x = 0\) does not exist in our equation. Hence, our function has no y-intercept. Now, setting \(y = 0\) gives \(1 - 3/x^{2} = 0\), which simplifies to \(x = \pm\sqrt{3}\). Thus, the x-intercepts are \(-\sqrt{3}\) and \(\sqrt{3}\).

Locate the asymptotes

Vertical asymptotes can be found where the denominator of a rational function is equal to zero. However, we see that \(x = 0\) is not in the domain of our function, implying a vertical asymptote at \(x = 0\). In addition, looking at the function as \(x\) grows very large in either the negative or positive direction, the function goes to 1. This tells us that the horizontal asymptote is \(y = 1\).

Finding the extrema

For this function, there are no maximum or minimum points because it is an asymptotically decaying function in both sides of x-axis, so it will keep declining but will never reach the value at y=1.

Plot the function

Now plot the function using all the identified properties. The graph will decline from a height near the y value of 1, intersect the x-axis at \(-\sqrt{3}\) and \(\sqrt{3}\), then ascend towards the horizontal asymptote of \(y = 1\), but it will never cross it. The vertical asymptote at \(x = 0\) is also a significant feature.

Key concepts

X-Intercepts of a Function

Understanding the x-intercepts of a function is crucial, as these are the points where the graph of the function crosses the x-axis. In essence, they represent the values of x for which the function's output, y, is zero. For the function y = 1 - 3x^-2, we look for x values that make y equal to zero. By setting y = 0 and solving the equation, we find two x-intercepts: \( x = +\sqrt{3} \) and \( x = -\sqrt{3} \). Visualizing this on a graph, these intercepts indicate where the curve will touch the x-axis at two distinct points.

Grasping the concept of x-intercepts helps students predict the behavior of the function around the x-axis, a fundamental aspect of graphing rational functions.

Vertical Asymptotes

When graphing rational functions, it's essential to understand vertical asymptotes. These are vertical lines to which the graph of a function approaches but never touches or crosses. A vertical asymptote occurs at values of x which make the function undefined. In our specific function, there is a vertical asymptote at \( x = 0 \), since it's not in the domain of the function, meaning the function will shoot off to infinity as it nears this x value from either side.

Determining vertical asymptotes lays the foundation for sketching accurate representations of rational functions, as they showcase critical boundaries within the graph.

Horizontal Asymptotes

The horizontal asymptotes of a function describe the behavior of the graph as x approaches large positive or negative values. For the function y = 1 - 3x^-2, the horizontal asymptote is y = 1. This line represents a 'ceiling' or 'floor' that the function's curve approaches but never quite touches as x moves towards infinity or negative infinity.

Knowing where the horizontal asymptote lies is a key factor in predicting the end behavior of a function and rounding out the overall shape of the graph in the long run. It completes the picture of how the function behaves at extreme values of x.

Extrema of a Function

The extrema of a function refer to the points on the graph where the function reaches its highest or lowest values—known as the maxima and minima, respectively. They represent important features in the function's overall shape. However, in the given function y = 1 - 3x^-2, there are no extrema, as the function is asymptotic and does not have any turning points where it reaches a maximum or minimum. As the function approaches the horizontal asymptote, it gets closer to y = 1 indefinitely but never actually attains this value.

Despite the absence of extrema in this case, identifying them in other functions is vital to understand the full range of a function's behavior and its graph's topography.

Asymptotic Behavior

The term asymptotic behavior refers to the way a graph approaches its asymptotes but doesn't actually make contact with them. For our example, y = 1 - 3x^-2, the function grows infinitely large as x approaches zero (the vertical asymptote), and it levels off near y = 1 and runs parallel to the x-axis at extreme values of x (the horizontal asymptote). This behavior gives a clear sense of directionality and helps in visual prediction, informing students how to depict the function's progression on a graph.

The concept of asymptotic behavior forms a critical component of understanding the long-term trends of functions, indispensable when tackling rational functions and their properties.

Function Sketching Techniques

Mastering function sketching techniques enables students to create accurate graphical representations of functions. The key steps usually involve finding x and y intercepts, locating horizontal and vertical asymptotes, determining the extrema, and examining the asymptotic behavior. Then, by plotting these critical points and considerations onto a coordinate grid, a skeleton of the graph evolves. Finally, by connecting these key points and keeping the function's behavior around the asymptotes in mind, the sketch can be completed.

This approach helps prevent misconceptions and allows for the detailed portrayal of the function's characteristics. For the given function y = 1 - 3x^-2, even though we have no y-intercepts or extrema, our graph is primarily guided by the horizontal asymptote, vertical asymptote, and x-intercepts for its complete representation.