The value of a new car in us dollars is modeled by the quadr

The value of a new car (in us dollars) is modeled by the quadratic function v(t) = 125t^2 - 3,000t + 22,000, where t represents the number of years since the car was purchased. Explain what the vertex of the function represents in the context.

Solution

We have v(t) = 125t² -3000t + 22000 = 125 ( t² -24t + 176) = 125 ( t² - 2*12t + 144) + 125*( 176-144) = 125(t -12)² + 4000 . This is the vertex form of the equation of a parabola with vertex at ( 12, 4000). The parabola opens upwards. It indicates that the price of the car keeps declining till 12 years after purchase when it reaches its minimum price ( $ 4000), which is after 12 years. Thereafter, the price of the car starts increasing once again.