Questions: Find the slant asymptote of the graph of the rational function. Follow the seven-step strategy and use the slant asymptote to graph the rational function. f(x)=(x^2+6 x-7)/(x-7) A. The x-intercept(s) is/are 1,-7. (Type an integer or a simplified fraction. Use a comma to separate answers if needed.) B. There are no x-intercepts. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The equation of the vertical asymptote is 7. (Type an equation.) B. There is no vertical asymptote.

Find the slant asymptote of the graph of the rational function. Follow the seven-step strategy and use the slant asymptote to graph the rational function. f(x)=(x^2+6 x-7)/(x-7) A. The x-intercept(s) is/are 1,-7. (Type an integer or a simplified fraction. Use a comma to separate answers if needed.) B. There are no x-intercepts. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The equation of the vertical asymptote is 7. (Type an equation.) B. There is no vertical asymptote.
Transcript text: Find the slant asymptote of the graph of the rational function. Follow the seven-step strategy and use the slant asymptote to graph the rational function. \[ f(x)=\frac{x^{2}+6 x-7}{x-7} \] A. The $x$-intercept(s) is/are $1,-7$. (Type an integer or a simplified fraction. Use a comma to separate answers if needed.) B. There are no x-intercepts. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The equation of the vertical asymptote is $\square$ 7. (Type an equation.) B. There is no vertical asymptote.
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Solution

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Solution Steps

Step 1: Find the slant asymptote

To find the slant asymptote, we perform polynomial long division on the rational function \( f(x) = \frac{x^2 + 6x - 7}{x - 7} \).

Step 2: Perform polynomial long division

\[ \begin{array}{r|rr} x - 7 & x^2 + 6x - 7 \\ \hline & x + 13 \\ \end{array} \] The quotient is \( x + 13 \), which is the slant asymptote.

Step 3: Find the vertical asymptote

The vertical asymptote occurs where the denominator is zero. Setting \( x - 7 = 0 \), we get: \[ x = 7 \]

Final Answer

The slant asymptote is \( y = x + 13 \).

The vertical asymptote is \( x = 7 \).

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 30}, "commands": ["y = (x^2 + 6x - 7)/(x - 7)", "y = x + 13"], "latex_expressions": ["$f(x) = \\frac{x^2 + 6x - 7}{x - 7}$", "$y = x + 13$"]}

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