The graph of a function of the form f(x)= a/x^2 +b is shown. Find the values of "a" and "b".

Question

A (m<n) graph is pictured with a clear Horizontal Asymptote on the X-Axis and no Vertical Asymptotes. The function runs through the points (-1,1),(0,2), &(1,1).

This is Q.28 in Unit 8.3 in the Larson Algebra 2 Textbook 2012 edition. (Green cover with track runners as the cover image.)

Answer 1

What we want to do with these sorts of questions is plug in x values and match them with the corresponding y values. Doing so will give us a system of equations, where we can then solve for the constants a and b.

f(-1) = a/((-1)² + b = 1

⇒ a/(1+b) = 1

⇒ a = 1 + b (cross multiplying)

⇒ a - b = 1 (subtracting b)

f(1) = a/((1)² + b) = 1

⇒ this is the same thing as f(-1) since (-1)²=(1)² = 1

⇒ a - b = 1

f(0) = a / ( 0² + b ) = 2

⇒ a / b = 2

⇒ a = 2b (multiplying by b)

There are several ways we can proceed with this but I think the most simple way would be to substitute a = 2b into a - b = 1 because we will immediately get b.

2b-b = 1

⇒ b = 1.

Since we know b = 1, a - 1 = 1

⇒ a = 2. (adding 1 to both sides)

So we have a = 2, b = 1.

Therefore, the function is 2/(x² + 1)

Hope this helps!