Write three functions. In the first function, y should vary directly with x. In the second function, y should vary inversely wit
h x. In the function, the relationship between x and y should be neither inverse variation nor direct variation. Describe the graph of each function and give a real world example for each.
<span>A) First function, y varies directly with x.
1) function: y = (3/4)x
2) graph: it is a straight line that passes through the origin and has slope 3/4. The slope means that the rate of change of the function is 3 units per every 4 units the x-value incresase or, what is the same 0.75 units per incresase unit of x - value.
3) real world example
A recipe of a cake instructs to use 3 cups of sugar for every 4 cups of flour. So, how much flour you need if you have 12 cups of sugar?
y = (3/4)x , so if x = 3, y = (3/4)*12 = 3*12/4 = 9.
So, given that the variation is direct you multiply the number of cups of sugar times the constant rate, 3/4, to get the number of cups of flour in relation with the given amount of sugar.
B) The second function: y varies inversely with x.
Inverse variation => y*x = constant or y = constant / x.
Tnat means that if x increase y will decrease in the same factor that x increases.
1) function: y = 12 / x
2) graph: the form of this graph is called hyperbola, it is a decreasing line from left to right. It has two asymptotes, the y-axis (x =0) and the x-axis (y = 0). That means that x and y can never be zero.
As the x-value approaches 0, the y value approaches positive or negative infinity; as the y-values approaches 0 the x-values approaches to positive or negative infinity.
If you take the positive values, the graph is a decreasing curve in the first quadrant (x and y are positive).
If you take the negative values, the graphs is a decreasing curve in the third quadrant (x and y are negative)
3) real world example.
The relatioship between velocity and time in a uniform motion.
If the distance run by an object is constant, as the velocity increases the time decreases in the same factor.
Suppse a distant of 100 km between cities A and B.
How long will it take to travel from A to B at 50 km/ h and 25 km/h ?:
100 km = velocity * time
at 50 km/h: 100 km = 50 km/h * t => t = [100 km ] / [50 km/h] = 2 hours
at 25 km/h: 100 kg = 25 km/h * t => t = [ 100 km ] / [25 km/h] = 4 hours.
C) Third case, the relationship between
x and y should is neither inverse variation nor direct variation.
Of course, there are infinite type of functions that are neither inverse variation nor direct variation: linear (that do not passe through the origin), quadratic, exponential, logarithmical, trigonometric sine, ...
1) example of function: y = 30 + 2x
2) graph: it is a straigh line with y-intercept 30 and slope 2.
3) real world example:
The cost of producing chairs consists of 30 dollars of rent for the facility plus 2 dollar to produce each chair, so the total cost y is 30 + 2x. </span>
