Problem 29 Sketch a graph of the function s... [FREE SOLUTION]

Chapter 0: Problem 29

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=\frac{3 x}{\sqrt{x^{2}+4}}$$

Short Answer

Expert verified

The function intercepts the origin (0,0), has no extreme points or vertical asymptote, and has horizontal asymptotes at lines y=3 and y=-3.

Step by step solution

Understand the Function

The given function is $ f(x) = \frac{3 x}{\sqrt{x^{2}+4}} $. This is a rational function, a ratio where both the numerator and denominator are polynomials, where the denominator is the square root of a quadratic equation.

Find the Intercepts

The x intercept is found by setting $ f(x) = 0 $. This implies that $ x = 0 $. The y intercept is found by setting $ x = 0 $. So, $ y = f(0) = 0 $. Therefore, the function intercepts the origin (0,0) .

Find the Extreme Points

The extreme points for this function are at the critical points or undefined points. Therefore, we find the derivative $ f'(x) = \frac{3[x^{2}+4-(2x^{2})]}{(x^{2}+4)^{2}} $. Solving $ f'(x) = 0 $ gives no real solutions - this function does not have any extreme points.

Find the Asymptotes

The vertical asymptote is the value of x where the function does not exist or goes to 卤 infinity. But, the function exists for all x values, therefore it does not have any vertical asymptotes.However, to find the horizontal asymptote, we need to take the limit of the function as x approaches to positive and negative infinity. Both of these limits give us y=3. $ \lim_{{x \to \infty}} f(x) = 3 $ and $ \lim_{{x \to -\infty}} f(x) = -3 $, therefore this function has a horizontal asymptote at y=3 and y=-3.

Sketch the Graph

Now, plot the intercept, the limit values, and draw curves connecting these values and the limit lines at y=3 and y=-3. This will help sketch the graph of the function.

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Key Concepts

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

Extreme Points

When discussing extreme points in a graph of a rational function, we're talking about local minima or maxima. These are points where the function reaches a lowest or highest value in its immediate vicinity. For our function, we found that solving the derivative $ f'(x) = 0 $ resulted in no real solutions. This means there are no extreme points in the graph.
The derivative simplifies as follows:

Calculate $ f'(x) = \frac{3(x^{2} + 4) - 6x^{2}}{(x^{2} + 4)^{2}} $.
This reduces to $ \frac{3 \times 4 - 3x^{2}}{(x^{2} + 4)^{2}} $.
Setting $ 12 - 3x^{2} = 0 $ yields no solution for real x.

Thus, the lack of extreme points suggests that the function steadily increases or decreases across its domain without reaching a peak or trough. This characteristic can be helpful when visualizing the behavior of the function along the x-axis.

Intercepts

Finding intercepts is a fundamental step when graphing rational functions. These intercepts tell us where the graph crosses the x-axis and y-axis.
For the function $ f(x) = \frac{3x}{\sqrt{x^{2}+4}} $:

X-intercept: To find the x-intercept, set $ f(x) = 0 $. Here, the only solution is $ x = 0 $, as zero in the numerator makes the whole function zero.
Y-intercept: For the y-intercept, set $ x = 0 $. Substituting this into the function gives $ f(0) = 0 $. Thus, the only point of intersection here is the origin, (0,0).

Getting these intercepts allows us to outline important "starting points" when sketching the graph. It anchors the visualization and aids in showing major changes in direction or crossing on the axes.

Asymptotes

Asymptotes in the graph of a rational function represent invisible boundaries the function approaches but never quite reaches. In this exercise, there are no vertical asymptotes because the function exists for all real values of $ x $; there are no x-values that make the denominator zero.
However, horizontal asymptotes can be identified by examining the values the function approaches as $ x $ moves towards positive or negative infinity.

Horizontal Asymptote: The horizontal asymptotes are found by calculating the limits $ \lim_{{x \to \infty}} f(x) = 3 $ and $ \lim_{{x \to -\infty}} f(x) = -3 $. So, the function approaches two horizontal lines: $ y = 3 $ and $ y = -3 $.

Including these asymptotes in the graph helps determine function behavior at the extremes of its domain, providing insight into how it stabilizes as the x-values become very large or very small.

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91影视

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=\frac{3 x}{\sqrt{x^{2}+4}}$$

Short Answer

Step by step solution

Understand the Function

Find the Intercepts

Find the Extreme Points

Find the Asymptotes

Sketch the Graph

Key Concepts

Extreme Points

Intercepts

Asymptotes

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