Write a piecwise function for the graph If you are a member

-Write a piecwise function for the graph

If you are a member of Powerhouse Gym, you can rent a locker by the hour. The cost is $2.00 per hour.

a.      What would the total cost be to rent a locker for 3 hours and 15 minutes?

b.       Write an equation for this function.

( It will look like this, y = m[x] )

Solution

As per the given graph, in the interval [-9, -6], the function is a line which passes through the points (-9,8) and (-6,-1). Let the equation of this line be y = mx + c where m is its slope and c is the y-intercept Then , m = (-1-8)/(-6+9) = -9/3 = -3. Also, on substituting m = -3, x = -6 and y = -1 in the equation, we get -1= -3*(-6)+c or, -1 = 18+c so that c = -1-18 = -19. Therefore, the equation of this line is y = -3x-19.

In the interval [-6,2], the function is a line which passes through the points (-6,-1) and (2,3). Let the equation of this line be y = mx + c where m is its slope and c is the y-intercept( where x = 0). Then, c=2 and m = (3+1)/(2+6) = 4/8 = ½. Therefore, the equation of this line is y = x/2+2.

In the interval [2,7], the function is a line which passes through the points (2,3) and (7,-2). Let the equation of this line be y = mx + c where m is its slope and c is the y-intercept. Then m = (-2-3)/(7-2) = -5/5 = -1. Also, on substituting m = -1, x = 2 and y = 3 in the equation, we get 3 = -1*2+c or, 3 = c-2 so that c = 3+2 = 5.Therefore, the equation of this line is y = -x+5.

Then, the equation of the piecewise function is

y = -3x-19 if x [-9, -6]

y = x/2+2, if x [-6,2]

y = -x+5 if x [2,7]

At the points of intersection of these 3 lines, i.e where x = -6, or, x = 2, the values of y will be common for both the intersecting lines as there are no holes.

(a). The cost of the locker for 3 hours, 15 minutes, for a non-member of Powerhouse Gym ( In case the graph represents this) will be the cost for 2 hours + cost for 1 hour 15 minutes (5/4 hour). The cost for first 2 hours will be 2/2+2 = 1+2 = $3 ( obtained by substitution in the equation of the 2^nd line). The cost for the next 5/4 hour will be -5/4 +5 = 15/4 = $ 3.75 ( obtained by substitution in the equation of the 3^rd line).Thus, the total cost of renting a locker for 3 hours, 15 minutes, for a non-member of Powerhouse Gym will be $ 3+$3.75 = $ 6.75. For a member of Powerhouse Gym, this cost will be $ 2*(13/4) = $13/2 = $6.50 . These 2 costs for non-members and also members are correct if the lockers can be rented for a fraction of an hour also. Otherwise, after 2 hours, the charges of another 2 hours will apply.

b. For a member, the equation is y = 2x , where x is the number of hours for which the locker is rented, and y is the rent of the locker for the period. For a non-member, the equation is y = x/2+2, if x 2 and y =   2(2/2)+2 + (-x)+5 = 4-x+5 or, -x+9 if x > 2.