A quadratic model curves rather than changing at a constant rate. If the question asks when its output is at least a target, convert the target to the model’s units, then graph the model and a horizontal line at that value in Desmos. Keep the inputs where the model is on or above the line, and check whether the question requires integer inputs.
Hints
- Hint 1
The model gives in thousands of dollars, so convert the dollar target to thousands before comparing it with . What number should the model’s output reach?
- Hint 2
A parabola is the curved graph of a quadratic. Graph the car’s value and a horizontal line at the target. Their crossings mark the boundaries, but the requested integer years may start and end inside them.
Step-by-step
Graph the target value
Step 1Translate the dollar target
Because is measured in thousands of dollars, $28,000 corresponds to . At least includes that value and anything greater, so the requested times satisfy:
- Step 2
Find where the model reaches the target
Type and , using for the model’s . Click both intersections, where the graphs meet. Desmos shows about and . These are the boundaries where the car is worth exactly $28,000.
- Step 3
Identify the qualifying part of the graph
The negative coefficient makes the parabola open downward. Between the two crossings, its graph is on or above the line, so those times meet the value target. Outside the crossings, the car’s modeled value is below it.
- Step 4
Keep only integer years
The question asks for integer values of . The first integer between and is , and the last is . So the car is worth at least $28,000 for , representing 2008 through 2012. Choice D.