Question 1111985
the function you provided is:


T(t) = .7 * t^2 + 4.2 * t + 98.6


it will not give you a maximum temperature.


this is because the coefficient of the t^2 term is positive.


in order for the function to give you a maximum temperature, the coefficint of the t^2 term has to be negative.


to graph this function:


let x = t and let y = T(t).


the function becomes y = .7 * x^2 + 4.2 * x + 98.6


the domain is 0 <= x <= 6.


here's the graph of the function as given.


<img src = "http://theo.x10hosting.com/2018/031301.jpg" alt="$$$" >


you can see that there is no maximum.


there is only a minimum.


here is the graph of the function when the coefficient of the x^2 term is negative.


<img src = "http://theo.x10hosting.com/2018/031302.jpg" alt="$$$" >

now you have a maximum.


from the graph, you can see that, when x = 0, the temperature is 98.6, and when x = 3, the temperature is 104.9, and when x = 6, the temperature is 98.6 again.


the maximum temperature is when x = 3.


the maximum temperature is 104.9 degrees.


we allowed y to be equal to T(t) and we allowed x to be equal to t in order for the graphing software to be able to handle the function.


some graphing functions can handle it the way it was originally written, but a lot can't, which is why we made the equivalency.


bottom line:


coefficient of the x^2 term, or the t^2 term, needs to be negative in order to get a maximum out of this quadratic function.


since the quadratic equation is in standard form, you could also find the maximum vaalue by using the formula:


x = -b/2a


the standard form of a quadratic equation is y = ax^2 + bx+ c, when y is equal to 0.


in this form:


a = the coefficient of the x^2 term.
b = the coefficient of the x term.
c = the constant term.


your equation of y = -.7 * x^2 + 4.2 * x + 98.6 gets you:


a = -.7
b = 4.2
c = 98.6


x = -b/2a becomes x = -4.2 / -1.4 which gets you x = 3.


when x = 3, y = -.7 * 3^2 + 4.2 * 3 + 98.6.


this results in y = 104.9


this agrees with what the graph is telling you, as it should.