14. Sketch a graph of the rational function f(x)=(2x+4)/(x^2)+7x+10 him (4) intercepts, y-intercepts, and asymptotes. holes, X-

To sketch the graph of the rational function $f(x)=\frac {2x+4}{x^{2}+7x+10}$, we need to find the x-intercepts, y-intercepts, and asymptotes.<br /><br />Step 1: Find the x-intercepts<br />The x-intercepts occur when the numerator of the rational function is equal to zero. So, we set the numerator equal to zero and solve for x:<br />$2x+4=0$<br />$2x=-4$<br />$x=-2$<br /><br />So, the x-intercept is at $x=-2$.<br /><br />Step 2: Find the y-intercepts<br />The y-intercepts occur when x is equal to zero. So, we substitute x=0 into the rational function and simplify:<br />$f(0)=\frac {2(0)+4}{(0)^{2}+7(0)+10}=\frac {4}{10}=\frac {2}{5}$<br /><br />So, the y-intercept is at $y=\frac {2}{5}$.<br /><br />Step 3: Find the asymptotes<br />The vertical asymptotes occur when the denominator of the rational function is equal to zero. So, we set the denominator equal to zero and solve for x:<br />$x^{2}+7x+10=0$<br />$(x+2)(x+5)=0$<br /><br />So, the vertical asymptotes are at $x=-2$ and $x=-5$.<br /><br />The horizontal asymptote is determined by the degrees of the numerator and denominator. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y=0$.<br /><br />Step 4: Sketch the graph<br />Now that we have found the x-intercepts, y-intercepts, and asymptotes, we can sketch the graph of the rational function.<br /><br />The graph will have a vertical asymptote at $x=-2$ and another vertical asymptote at $x=-5$. The graph will also have a horizontal asymptote at $y=0$. The graph will approach the asymptotes but never touch them.<br /><br />The graph will also pass through the x-intercept at $x=-2$ and the y-intercept at $y=\frac {2}{5}$.<br /><br />Final Answer: The graph of the rational function $f(x)=\frac {2x+4}{x^{2}+7x+10}$ has a vertical asymptote at $x=-2$, a vertical asymptote at $x=-5$, a horizontal asymptote at $y=0$, an x-intercept at $x=-2$, and a y-intercept at $y=\frac {2}{5}$.