A quadratic revenue model in vertex form shows where its graph turns. If the number multiplying the square is negative, the graph opens downward, so its highest point is the maximum. Graph the model in Desmos and click that point. Read the price from the input coordinate, not the revenue from the output coordinate.
Hints
- Hint 1
The vertex is where a parabola turns. A negative number multiplying a square makes the vertex a high point. After graphing the model, which point should you click?
- Hint 2
A point on this graph pairs a selling price with a revenue. The input is the price, and the output is the revenue. Which coordinate of the highest point answers the question?
Step-by-step
Approach 1: Graph the revenue model
Step 1Identify the maximum on the graph
The negative coefficient makes the graph of this quadratic a downward-opening parabola, a curve that turns once. Its vertex, or turning point, is its highest point.
- Step 2
Read the price at the highest point
Type , using in place of the price . Click the graph, then its highest point. Desmos shows . The first coordinate is the price; the second is the revenue. So a selling price of $40 maximizes the revenue. Choice B.
Approach 2: Use the square to find the peak
Step 1Find the revenue's upper limit
A square is never negative. So subtracting times a square from cannot make the revenue greater than : . The model reaches that limit when the squared part is zero.
- Step 2
Find the price that reaches the limit
Set the expression inside the square to zero so the revenue reaches its maximum: . Add to both sides: . Because is the selling price in dollars, the price that maximizes revenue is $40. Choice B.