# How do you use the important points to sketch the graph of f(x) = -x^2 + 14x - 48?

Apr 15, 2017

graph{-x^2 + 14x - 48 [-18.18, 27.43, -16.63, 6.17]}

#### Explanation:

Find maximum point or vertex:

$\max \left(a {x}^{2} + b x + c\right) = \left(\frac{- b}{2 a} , f \left(\frac{- b}{2 a}\right)\right)$

In this case: $b = 14 , a = - 1$

$\max \left(- {x}^{2} + 14 x - 48\right) = \left(\frac{- 14}{- 2} , f \left(\frac{- 14}{- 2}\right)\right)$
$\max \left(- {x}^{2} + 14 x - 48\right) = \left(7 , f \left(7\right)\right)$
$\max \left(- {x}^{2} + 14 x - 48\right) = \left(7 , 1\right)$

Now let's see if it opens up or downwards. As $a < 0$, it opens downwards.

Now let's see where it cuts both the x axis ($y = 0$) and the y axis ($x = 0$).

x-axis ($y = 0$): $- {x}^{2} + 14 x - 48 = 0 \implies x = 6 , 8$
y-axis ($x = 0$): $0 + 0 - 48 \implies y = - 48$

Plot the max point $\left(7 , 1\right)$, the two points on the x-axis the parabola intersects $\left(6 , 0\right) \text{ and } \left(8 , 0\right)$, the point in the y axis it intersects $\left(0 , - 48\right)$, and join them together smoothly .

Ta-da!