Question

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

Answer 1

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

Explanation:

Find maximum point or vertex:

`max(ax^2 + bx + c) = ((-b)/(2a), f((-b)/(2a)))`

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

`max(-x^2 + 14x - 48) = ((-14)/(-2), f((-14)/(-2)))`
`max(-x^2 + 14x - 48) = (7, f(7))`
`max(-x^2 + 14x - 48) = (7, 1)`

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 + 14x - 48=0 => x = 6,8`
y-axis (`x = 0`): `0 + 0 - 48 => y = -48`

Plot the max point `(7,1)`, the two points on the x-axis the parabola intersects `(6,0) " and " (8,0)`, the point in the y axis it intersects `(0,-48)`, and join them together smoothly .

Ta-da!