Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function $ f(x) = - {x^2} + 6x + 6? $

Answer 1

Hint: As we can see that the above equation of the form $ y = a{x^2} + bx + c $ . We know that it is the standard form of the equation of parabola. For parabola we decide whether it has maximum or minimum by observing the sign of the $ {x^2} $ coefficient. We can find the minimum and maximum value with the formula $ \dfrac{{ - b}}{{2a}} $ . So we will compare the given equation and the standard form and calculate the required value.

Complete step by step solution:
As per the question we have the equation $ f(x) = - {x^2} + 6x + 6 $ . We can observe that the coefficient of $ {x^2} $ is $ - 1 $ , so the given equation has maximum value.
The standard form of the equation is $ y = a{x^2} + bx + c $ , so by comparing the equations we have $ b = 6,a = - 1 $ and $ c = 6 $ .
So the maximum value of the given equation is
 $ \dfrac{{ - b}}{{2a}} = \dfrac{{ - 6}}{{2( - 1)}} $ . It gives us the value $ {x_{\max }} = 3 $ .
Now we can calculate the maximum value of $ y $ , by substituting the value of $ x $ . So we have $ {y_{\max }} = - {3^2} + 6 \times 3 + 6 $ , on further solving we have
 $ {y_{\max }} = - 9 + 18 + 6 = 15 $ .
Hence the maximum value of the given equation is $ \left( {3,15} \right) $ .
Since the coefficient of $ {x^2} $ is $ - 1 $ , so if the coefficient is negative we have the shape of the graph is that the parabola opens downward.

Note: We should note that if the coefficient of $ {x^2} $ is positive then we will calculate the maximum or minimum value of the equation by using the same formula and method. The graph of such an equation is that the parabola opens upward. We know that the axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves.