A function f is define f(x)=-x^2+x+2(a) express f in standard form&nbsp;(b) sketch the graph of f&nbsp;(c) find the maximum value of f&nbsp;

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2022-06-04

A function f is define f(x)=-x^2+x+2

(a) express f in standard form

(b) sketch the graph of f

karton

Expert2023-05-19Added 613 answers

(a) To express the function

f (x) = - x^{2} + x + 2

in standard form, we need to rewrite it in the form

f (x) = a x^{2} + b x + c

, where

a

b

, and

c

are constants.
By rearranging the terms, we have:

f (x) = - x^{2} + x + 2

f (x) = - 1 \cdot x^{2} + 1 \cdot x + 2

Therefore, the function

f (x)

in standard form is:

f (x) = - x^{2} + x + 2

.
(c) To find the maximum value of

f (x)

, we can use the vertex formula. The vertex of a quadratic function in the form

f (x) = a x^{2} + b x + c

is given by the coordinates

(- \frac{b}{2 a}, f (- \frac{b}{2 a}))

.
For the function

f (x) = - x^{2} + x + 2

, we have

a = - 1

b = 1

, and

c = 2

. Substituting these values into the vertex formula, we get:

x_{vertex} = - \frac{1}{2 (- 1)} = \frac{1}{2}

Substituting

x_{vertex}

into the function

f (x)

, we have:

f (\frac{1}{2}) = - {(\frac{1}{2})}^{2} + \frac{1}{2} + 2

Simplifying, we find:

f (\frac{1}{2}) = - \frac{1}{4} + \frac{1}{2} + 2 = \frac{7}{4}

Therefore, the maximum value of

f (x)

\frac{7}{4}

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