Question

Find a parabola with equation $ y = a{x^2} + bx + c $ that has slope \[4\] at $ x = 1 $ and slope $ \left( { - 8} \right) $ at $ x = - 1 $ and passes through $ \left( {2,15} \right) $ .

Answer 1

Hint: The given problem tests us on the concepts of applications of derivatives as well as conic sections. The problem requires us to find the parabola with equation $ a{x^2} + bx + c = 0 $ given certain conditions. We first have to analyze the problem by drawing a figure for the situation and then plan our sequence of actions. Such problems put our knowledge of all the fields of mathematics such as algebra, calculus, conic sections and analytical geometry to test.

Complete step by step solution:
Rough figure of parabola is shown below,

We are given the equation of parabola as $ y = a{x^2} + bx + c $ . Also, slope of parabola at $ x = 1 $ is $ 4 $ and at $ x = - 1 $ is $ - 8 $ .
Hence, Slope of parabola $ = \dfrac{{dy}}{{dx}} $
 $ = \dfrac{{d\left( {a{x^2} + bx + c} \right)}}{{dx}} $
 $ = 2ax + b $
So, slope of parabola at $ x = 1 $ is $ 2a\left( 1 \right) + b = 2a + b $
 $ = 2a + b = 4 - - - - - - - (1) $
Also, slope of parabola at $ x = - 1 $ is $ 2a\left( { - 1} \right) + b = - 2a + b $
 $ = - 2a + b = - 8 - - - - - - (2) $
Adding both the equations $ (1) $ and $ (2) $ , we get,
 $ 2b = - 4 $
 $ = b = - 2 $
Putting value of b in equation $ (1) $ to obtain value of a.
We get, $ 2a - 2 = 4 $
 $ = 2a = 6 $
 $ = a = 3 $
So the value of a is $ 3 $ and value of b is $ - 2 $ .
So, we get the equation of parabola as $ y = 3{x^2} - 2x + c $ .
Now, the parabola passes through $ (2,15) $ ,
Hence, point $ (2,15) $ lies on the parabola.
So putting $ x = 2 $ and $ y = 15 $ in the equation of parabola,
 $ 15 = 3{\left( 2 \right)^2} - 2\left( 2 \right) + c $
 $ = 15 = 12 - 4 + c $
 $ = c = 7 $
So, the value of c is $ 7 $ .
Hence, the equation of parabola is $ y = 3{x^2} - 2x + 7 $ .
So, the correct answer is “ $ y = 3{x^2} - 2x + 7 $ ”.

Note: The given question is a typical problem of applications of derivatives and also involves basic understanding of equations of conic sections such as parabola. Such problems illustrate the interdependence of mathematical ideas and topics on each other. Care should be taken while finding the slope of curves at a given point