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Using Transformations to Graph a Rational Function · Physics Playground

Exercises Polynomials and factoring

Example 2 from Algebra and Trigonometry, 5.6 Rational Functions

Using Transformations to Graph a Rational Function

Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Shifting the graph left 2 and up 3 would result in the function

$f (x) = \frac{1}{x + 2} + 3$

or equivalently, by giving the terms a common denominator,

$f (x) = \frac{3 x + 7}{x + 2}$
The graph of the shifted function is displayed in Figure 7.

Figure 7

Notice that this function is undefined at $x = −2,$ and the graph also is showing a vertical asymptote at $x = −2.$
$As x \to - 2^{-}, f (x) \to - \infty, and as x \to - 2^{+}, f (x) \to \infty .$

As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at $y = 3.$

$As x \to \pm \infty, f (x) \to 3.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.

How did it go?

PreviousUsing Arrow Notation Next Solving an Applied Problem Involving a Rational Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

Your turn