### About Graphing Rational Functions:

Our goal for this section is to be able to create a rough sketch of the graph of a rational function. In the lesson, we showed how to graph transformations of y = 1/x and y = 1/x^{2}. For rational functions that can't be sketched using transformations, we gave a simple procedure. First, we will sketch all asymptotes. Second, we will find and label the x and y intercepts. Third, we will find and label any point of intersection with the nonvertical asymptote. Fourth, we will split the number line up into intervals based on the x-intercepts and vertical asymptotes. Then, we will find and plot a point in each interval. Lastly, we will create a rough sketch of the graph.

Test Objectives

- Demonstrate the ability to graph transformations of y = 1/x and y = 1/x
^{2} - Demonstrate the ability to find asymptotes, holes, and intercepts for the graph of a rational function
- Demonstrate the ability to create a rough sketch of the graph of a rational function

#1:

Instructions: Give the asymptotes and intercepts, and then sketch the graph.

$$a)\hspace{.2em} f(x) = \frac{2}{x + 1} + 1$$

$$b)\hspace{.2em} f(x) = \frac{-2x - 2}{x + 6}$$

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#2:

Instructions: Give the asymptotes and intercepts, and then sketch the graph.

$$a)\hspace{.2em} f(x) = \frac{4}{\left(\frac{1}{3}x - 2\right)^2} - 5$$

$$b)\hspace{.2em} f(x) = -\frac{2}{(x - 3)^2} + 1$$

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#3:

Instructions: Give the asymptotes and intercepts, and then sketch the graph.

$$a)\hspace{.2em} f(x) = \frac{x^3 - x^2}{x - 1}$$

$$b)\hspace{.2em} f(x) = \frac{2x^3 - 3x^2 + x}{x^2 - \frac{3}{2}x + \frac{1}{2}}$$

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#4:

Instructions: Give the asymptotes and intercepts, and then sketch the graph.

$$a)\hspace{.2em} f(x) = \frac{2x^2}{x^2 + x - 6}$$

$$b)\hspace{.2em} f(x) = \frac{x^2 - 3x + 2}{4x + 4}$$

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#5:

Instructions: Give the asymptotes and intercepts, and then sketch the graph.

$$a)\hspace{.2em} f(x) = \frac{x^2 - 2x - 15}{(x - 1)(x - 2)(x + 3)}$$

$$b)\hspace{.2em} f(x) = \frac{(x + 2)^2(x - 1)}{(x-5)^2}$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ Vertical Asymptote: $$x = -1$$ Horizonal Asymptote: $$y = 1$$ x-intercept: $$(-3, 0)$$ y-intercept: $$(0, 3)$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ Vertical Asymptote: $$x = -6$$ Horizonal Asymptote: $$y = -2$$ x-intercept: $$(-1, 0)$$ y-intercept: $$\left(0, -\frac{1}{3}\right)$$ Desmos Link for More Detail

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#2:

Solutions:

$$a)\hspace{.2em}$$ Vertical Asymptote: $$x = 6$$ Horizonal Asymptote: $$y = -5$$ x-intercepts: $$\left(\frac{6\sqrt{5}}{5} + 6, 0\right), \left(-\frac{6\sqrt{5}}{5} + 6, 0\right)$$ y-intercept: $$\left(0, -4\right)$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ Vertical Asymptote: $$x = 3$$ Horizonal Asymptote: $$y = 1$$ x-intercepts: $$\left(\sqrt{2} + 3, 0\right), \left(-\sqrt{2} + 3, 0\right)$$ y-intercept: $$\left(0, \frac{7}{9}\right)$$ Desmos Link for More Detail

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#3:

Solutions:

$$a)\hspace{.2em}$$ Asymptotes: $$\text{None}$$ x-intercept: $$(0, 0)$$ y-intercept: $$(0, 0)$$ Desmos Link for More Detail Note: If you are using Desmos, you need to mouse over where x is 1 to see the hole in the graph.

$$b)\hspace{.2em}$$ Asymptotes: $$\text{None}$$ x-intercept: $$(0, 0)$$ y-intercept: $$(0, 0)$$ Desmos Link for More Detail Note: If you are using Desmos, you need to mouse over where x is 1/2 and 1 to see the holes in the graph.

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#4:

Solutions:

$$a)\hspace{.2em}$$ Vertical Asymptotes: $$x = -3, 2$$ Horizonal Asymptote: $$y = 2$$ x-intercept: $$\left(0, 0\right)$$ y-intercept: $$\left(0, 0\right)$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ Vertical Asymptote: $$x = -1$$ Oblique (Slant) Asymptote: $$y = \frac{1}{4}x - 1$$ x-intercepts: $$\left(2, 0\right), \left(1, 0\right)$$ y-intercept: $$\left(0, \frac{1}{2}\right)$$ Desmos Link for More Detail

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#5:

Solutions:

$$a)\hspace{.2em}$$ Vertical Asymptotes: $$x = 1, 2$$ Horizonal Asymptote: $$y = 0$$ x-intercept: $$\left(5, 0\right)$$ y-intercept: $$\left(0, -\frac{5}{2}\right)$$ Desmos Link for More Detail Note: If you are using Desmos, you need to mouse over where x is -3 to see the hole in the graph.

$$b)\hspace{.2em}$$ Vertical Asymptote: $$x = 5$$ Oblique (Slant) Asymptote: $$y = x + 13$$ x-intercepts: $$\left(-2, 0\right), \left(1, 0\right)$$ y-intercept: $$\left(0, -\frac{4}{25}\right)$$ Desmos Link for More Detail