Write your own real-world rational function. Use information from the web to support your rational function, graph it, and include a table of data.

Question

1. Write your own real-world rational function. Use information from the web to support your rational function, graph it, and include a table of data.

Answer 1

I have chosen the rational function y = f(x) = (x-2)/(2x²-7x+3).

Let's try to factor the denominator using the quadratic formula.

2x²-7x+3 = ax²+bx+c,

so a = 2, b = -7, and c = 3.

The values of x that satisfy 2x²-7x+3 = 0 are

x = (-b±(b²-4ac)^1/2)/2a

= (7±(7²-4*2*3)^1/2)/2*2

= (7±(49-24)^1/2)/4

= (7±(25)^1/2)/4

= (7±5)/4

Using the plus sign,

x = (7+5)/4

= 12/4

= 3

Using the minus sign,

x = (7-5)/4

= 2/4

= 1/2

Therefore, the denominator is 2x²-7x+3 = 2(x - 1/2)(x - 3) = (2x - 1)(x - 3).

The denominator is equal to 0 at x = 3 and x = 1/2, so that's where the vertical asymptotes are.

The domain of this function is all real numbers except the points x = 3 and x = 1/2.

There is on horizontal asymptote, at y = 0.

To prove this, let x be a large positive number, like 1000.

Then x² = 1,000,000,

y = f(1000) = (1000-2)/(2*1,000,000-7*1000+3) approximately = 1000/(2*1,000,000) = 1/2000.

Now let x be a large negative number, like -1000.

Then x² = 1,000,000,

y = f(-1000) = (-1000-2)/(2*1,000,000+7*1000+3) approximately = -1000/(2*1,000,000) = -1/2000.

So does an asymptote at y = 0 mean that the range is all real numbers except 0?

No, because at x=2, y = f(2) = (2-2)/(2*2²-7*2+3) = 0, so the range is all real numbers.

For a table of values and a graph, see the video answer to this question.

The table of values and the graph were created in Microsoft Excel.

The graph type in Excel is "X Y (Scatter)", the subtype is "Scatter with Smooth Lines and Markers".