Write three functions. in the first function, y should vary directly with x. in the second function, y should vary inversely with 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.

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.

2x^2 -2x -12 = 02(x -3)(x +2) = 0see below for explanation

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.