Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=3+\frac{2}{x} $$

Short answer

Vertical Asymptote: \( x = 0 \), Horizontal Asymptote: \( y = 3 \), X-intercept: \( x = -\frac{2}{3} \), Function is not symmetric and the graph is a hyperbola shifted up three units in the y-direction.

Step-by-step solution

Identify Vertical Asymptotes

Firstly, determine the value of \( x \) for which the denominator equals to zero, as these points will be undefined in the given function. The denominator is \( x \). So, setting \( x = 0 \), results in undefined value. Hence, \( x = 0 \) is the vertical asymptote.

Identify Horizontal Asymptotes

For \( x \) approaching \( \pm \) infinity, observe the behavior of function. It can be seen that for large values of \( x \), the fraction \( \frac{2}{x} \) approaches zero. Thus, \( y = 3 \) is the horizontal asymptote.

Find Intercepts

Intercepts are points where the graph crosses the x and y axes. For y-intercept, set \( x = 0 \), but here this results in undefined value. Thus there's no y-intercept. However, for x-intercept, set \( y = 0 \) and solve for \( x \). You are left with \( x = \frac{2}{-3} \), thus the x-intercept is \( -\frac{2}{3} \).

Check for Symmetry

A function is symmetric about the y-axis if its graph remains unchanged when \( x \) is replaced with \( -x \). If we replace \( x \) by \( -x \) in the given equation, we do not get the original function. Therefore, the function is not symmetric about the y-axis.

Sketch the Graph

Now, plot the asymptotes and intercept. The graph of \( y = 3 + \frac{2}{x} \) would be a hyperbola shifted three units in the positive y-direction. Remember to check result using a graphing utility to verify this.