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the profit P (in million of dollars) for a recreational vehicle retailer is modeled by a quadratic function...

on the form P=at^2+bt+c, where t represents the year. If you were president of the company, which of the following models would you prefer? Explain all plz.
(a) a is positive and t>=-b/(2a)
(b) a is positive and t<=-b/(2a)
(c) a is negative and t>=-b/(2a)
(a) a is negative and t<=-b/(2a)

Public Comments

1. First of all, you must be familiar with the quadratic graph, which resembles a parabola. The quadratic function is ax^2 + bx + c = 0 where a is not equal to 0.

You must know that when a>0 , the parabola is kinda like a U shape, while when a<0, the graph is inverted U-shape.

Now, you also must know that -b/2a is the equation of the line of SYMMETRY of the graph. If you were to draw a graph of y = x^2, the line of symmetry would be x=0, right?

Now, on to the question. First, I would like you to sketch a curve of P against t. You don't need graph paper, as the explanations below are purely qualitative.

I would choose model (a), because, if we were to draw this graph, we would get a U-shaped curve. Now, point your finger at where the line of symmetry would be. Move your finger to the right. What you are doing is INCREASING the value of t. As you increase the value of t, what happens to the value of P? You can see from the graph that the curve keeps on sloping upwards. This means that, as you increase t, P also increases rapidly.

What any business wants is increasing profits, right? So, that's why I chose model (a).

Model (b) also has the same shaped curve, if we analyze this graph, we find that, as t DECREASES, P increases. The only way t would decrease in this question, is that if we were to go back in time!

Model (c) and (d) both have inverted U-shape. If you move your finger to left or to the right, the curve slopes downwards, meaning that if you increase or decrease t, your profits will decrease. Who wants their profits to decrease?!

So, model (a) is my answer. Hope my explanation was helpful =)