 The Problem The quadratic function given by models the number of AIDS deaths from 1984 to 1994. Determine an equation whose solution is the year when the number of AIDS deaths might reach 500,000. Then solve the equation graphically and symbolically. The number 500,000 in this problem represents the value of  f(x). So replacing f(x) with 500,000 gives us the following equation to solve: ## Solving the Equation Graphically

To solve this equation graphically, begin by pressing and then enter 2375x2+5134x+5020 for Y1 and enter 500000 for Y2 as shown below. Now we must select and set an appropriate Viewing RectangleThe Viewing Rectangle of
[-20, 20, 10] by [400000, 600000, 10000] will be used.  To set this Viewing Rectangle,  press WINDOW and then enter -40 for Xmin, 40 for Xmax, 10 for Xscl, -500000 for Ymin, 500000 for Ymax, and 10000 for Yscl.

Press GRAPH to see the graph of the function as shown below.  Next find the intersections of the graphs. However, in this problem, it only makes sense to find the positive x-intercept since a year cannot be negative. Press 2nd TRACE[CALC], then press 5 to select Intersect. The results are shown below.  While the cursor mark is not visible, but it is flashing somewhere on the parabola. Press ENTER. Since the calculator now asks for "Second curve?", use the right arrow key to move the cursor close to this intersection point, and then press ENTER.  Now press ENTER once more. The calculator screens should appear as shown below.    The last screen shows the Intersection:   X 13.4     Y = 500000. Thus  X 13.4 is the desired solution for a future year.

Since x = 13 corresponds to 1984 + 13 = 1997. Thus, the number of AIDS deaths might reach 500,000 people in 1997.