Given the rational function. f(x)= \frac {3x+4}{-5x+8}. Determine the following: a. Where the...


Given the rational function. {eq}f(x)= \frac {3x+4}{-5x+8} {/eq}. Determine the following:

a. Where the vertical asymptote occurs?

b. At what point {eq}x {/eq} intercept occurs?

c. At what point {eq}y {/eq} intercept occurs?

d. Where the non vertical asymptote occurs?

e. Draw the graph for the given function.

Properties of rational functions

This example illustrates how to determine various properties of the graph of a rational polynomial function.

The example shows how to determine the intercepts with the coordinate axes, plus the equations of any asymptotes.

Answer and Explanation:

Part a

The vertical asymptote occurs when the denominator is zero.


{eq}-5x + 8 = 0 \\ \Rightarrow x = \dfrac{8}{5}. {/eq}

Part b

The x intercept occurs when y = 0.


{eq}\dfrac {3x+4}{-5x+8} = 0 \\ \Rightarrow 3x + 4 = 0 \\ \Rightarrow x = -\dfrac{4}{3}. {/eq}

So the x intercept is at {eq}(-\dfrac{4}{3}, 0). {/eq}

Part c

The y intercept occurs when x = 0.

This gives:

{eq}y= \dfrac {3(0)+4}{-5(0)+8} = \dfrac{1}{2}. {/eq}

So the y intercept is at {eq}(0, \dfrac{1}{2}). {/eq}

Part d

The non-vertical asymptote can be found by considering the highest powers of x in the numerator and denominator.

So, as {eq}\displaystyle x_{\to \infty}, {/eq} {eq}\dfrac {3x+4}{-5x+8} \to \dfrac{3x}{-5x} = -\dfrac{3}{5}. {/eq}

Hence the non-vertical asymptote is {eq}y = -\dfrac{3}{5}. {/eq}

Part d

Below is a sketch of the graph of the function:

Learn more about this topic:

Rational Function: Definition, Equation & Examples

from GMAT Prep: Help and Review

Chapter 10 / Lesson 11

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