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: or by subtracting 500,000 from both sides:

## Solving the Equation Graphically

To solve this equation graphically, begin by pressing and then enter for as shown below.

Now we must select and set an appropriate Viewing RectangleThe Viewing Rectangle of
[-40, 40, 10] by [-500000, 500000, 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 zeroes or x-intercepts of the graph. 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 2 to select zero. The results are shown below.

While the cursor mark is hard to see, it is flashing at the vertex of the parabola.   Use the right arrow key to move the cursor so that it is slightly to left of the point where the graph crosses the positive x-axis, and then press ENTER. Since the calculator now asks for "Right Bound", use the right arrow key to move the cursor slightly to the right of this same zero, 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 = 0. Thus  X 13.4 is the zero desired.

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