An inverse variation is a relation between two variables such that one variable increases, the other decreases proportionally. For the table to represent an inverse variation, the product of x and y must be constant. Find the product, xy, for each column in the table.
Find the product.
When a relation between x and y is an inverse variation, we say that x varies inversely as y. Inverse variation is modeled by the equation y = k/x or with an equivalent form x = k/y or xy = k, when k ≠ 0. The variable k represents the constant of variation, the number that relates the two variables in an inverse variation.
In this table, the constant of variation is 24.
On a guitar, the string length, s, varies inversely with the frequency, f, of its vibrations.
The frequency of a 26-inch E-string is 329.63 cycles per second. What is the frequency when the string length is 13 inches?
Solution:
s = k/f————————–Write the equation for an inverse variation.
26 = k/329.63—————-Substitute 26 and 329.63 for s and f.
8570.38 = k ——————-Multiply by 329.63 to solve for k.
After solving for k, write an equation for an inverse variation.
s = 8570.38/f—————–Substitute 8570.38 for k in the equation.
13 = 8570.38/f—————Substitute 13 for s in the equation.
f = 659.26 ———————Solve for f.
So, the frequency of the 13-inch string is 659.26 cycles per second.
How do you graph the reciprocal function, y = 1/x?
The reciprocal function maps every non-zero real number to its reciprocal.
Step 1: Consider the domain and range of the function.
Step 2: Graph the function.
Step 3:
Observe the graph of y = 1/x as it approaches positive infinity and negative infinity.
An asymptote is a line that a graph approaches. Asymptotes guide the end behaviour of a function.
As x approaches infinity, f(x) approaches 0. The same is true as x-values approach negative infinity, so the line y = 0 is a horizontal asymptote.
Step 4:
Observe the graph of y = 1/x as x approaches 0 for positive and negative x-values.
For positive values of x, as x approaches 0, f(x) approaches positive infinity.
For negative values of x, as x approaches 0, f(x) approaches negative infinity. The domain of the function excludes 0, so the graph will never touch the line x = 0. The line x = 0 is a vertical asymptote.
Graph g(x) = (1/x – 3)+ 2. What are the equations of the asymptotes? What are the domain and the range?
Start with the graph of the parent function,
Recall that adding h to x in the definition of f translates the graph of f horizontally. Adding k to f(x) translates the graph of f vertically.
The function (1/x – 3) + 2 is a transformation of the parent function f that shifts the graph of f horizontally by h units and then shifts the graph of f vertically by k units.
The graph of g(x) = (1/x – 3) + 2 is a translation of the graph of the parent function 3 units right and 2 units up.
The line x = 3 is a vertical asymptote. The line y = 2 is a horizontal asymptote.
Question 1
Determine if this table of values represents an inverse variation.
Solution:
Since the product xy is constant, this table of values represents an inverse variation.
Question 2
The amount of time it takes for an ice cube to melt varies inversely to the air temperature, degrees. At 20° Celsius, the ice will melt in 20 minutes. How long will it take the ice to melt if the temperature is 30° Celsius?
Solution:
Let t be the time it takes for an ice cube to melt and T be the air temperature in degrees.
t = k/T (Inverse Variation)
For T = 20 degrees, t = 20 minutes.
Equation: t = 400/T
If T = 30 degrees, t = 400/30 = 13.33 minutes.
Question 3
Graph g(x) = (1/x+2) – 4. What are the equations of the asymptotes? What are the domain and the range?
Solution:
Start with the graph of the parent function, g(x) = 1/x
Adding 2 to x shifts the graph horizontally to the left by 2 units and subtracting 4 from the output shifts the graph down by 4 units.
Below is the graph of the function: g(x) = (1/x+2) – 4
As we can see in the graph, vertical asymptote is x = -2 and horizontal asymptote is y = -4.
Graph the following functions. What are the asymptotes? What are the domain and the range?
