Question

Shannon manages a small zoo and she has been analyzing the attendance data. Shannon finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 3x. Shannon also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by f(x) = −x + 4. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.
On the graph two functions are joined at (1,3) but I would like to know how to solve it with the visual graph.

Answer 1

Answer:

The solution to this set of functions is (1,3) or x=1 and y=3. This solution represents the temperature at which the number of visitors is equal to the number of people who leave the park early.

Since it is graphed, and by the basic rules of functions, we know that the intersection of these two functions will be the answer. When you look at the graph, you can see the two functions intersect at (1,3). You can also check if this is correct by substituting these values back into these functions. For example, 3=3^1 is true and so is 3=-1+4, therefore, the solution is correct.

Answer 2

To graph the two functions, set arbitrary functions of x. As you set different values of x, you get their corresponding values of y. When you have both coordinates, you plot them as shown in the picture. The blue line represents the function f(x) = 3x, while the orange line represents the function f(x) = -x+4.

Indeed, the two lines intersect at point (1,3). To solve this analytically, set the system of equations:

y = 3x
y = -x + 4

Equating both equation of y,

3x = -x + 4
4x = 4
x = 1

Then, use any of the original equations: y = 3x = 3(1) = 3. Hence, the solution is (1,3). What does this solution mean? It means that when temperature is at 3 units, the people leaving and entering the small zoo are equal. Meaning, for every 1 person that comes in, another 1 leaves.