Question

Find a rational function for the graph provided.

 

Answer 1

 

• observe the graph,that  it crasses the x - axis at - 3 and 3.

From that, it must be some thing like : (x + 3)(x - 3) . a . n.

• Further inspection shows the x = 4 is a vertical asymptote.So, " a " must be like : 1/(x - 4).
• So, the graph should come from the equation { [(x + 3)(x - 3)]/(x - 4) } . n.
• The graph crosses the y - axis at (0, 5).

Means that, f (0) = 5

{ [(0 + 3)(0 - 3)]/(0 - 4) } . n = 5

(9/4)n = 5

⇒ n = 20/9.

• Substitute n = 20/9 in { [(x + 3)(x - 3)]/(x - 4) } . n.

= [ 20(x + 3)(x - 3) ]/[ 9(x - 4) ]

= [ 20(x ² - 9)] / [ 9(x - 4) ].

• The rational function is [ 20(x + 3)(x - 3) ] / [ 9(x - 4) ].

Answer 2

The general form of rational function f(x) = N(x) / D(x), where N(x) and D(x) are polynomials.

First find the polynomial N(x) :

The numerator of a rational function in lowest terms determines the x - intercepts of its graph. From the graph shown in Figure has x - intercepts - 3 (odd multiplicity since graph crosses the x - axis) and 3 (odd multiplicity, since graph crosses the x - axis).So one possibility for the numerator is N(x) = (x + 3)(x - 3).

Next find the polynomial D(x) :

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph. The vertical asymptotes of the graph is x = 4. Since f(x) approaches ∞ to the right of x = 4 and f(x) approaches - ∞ to the left of x = 4, we know that  is a factor of odd multiplicity in D(x). A possibility for the denominator is D(x) = (x - 4).

So, far we have f(x) = [(x - 3)(x + 3)] / [x - 4] = (x² - 9) / (x - 4).

To check the rational function, find the slant (or oblique) asymptote by using long division.

From the graph, the slant asymptote is y = 2x + 8 = 2(x + 4), but algebraically method slant asymptote is y = x + 4.

So, far we have f(x) = 2(x² - 9) / (x - 4).