For the quadratic function f(x) = -x^2 + bx + c, where b and c are positive constants, the… — SAT

For the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are positive constants, the maximum value of $f(x)$ is $k$ . The distance between the two $x$ -intercepts of the graph of $y = f(x)$ in the $xy$ -plane is equal to $\frac{2}{3}k$ . If the graph of $y = f(x)$ passes through the point $(1, 8)$ , what is the value of $b$ ?

Cevap: 4

Cevap

The correct answer is 4.

The correct answer is 4. By expressing the maximum value

k

and the distance between the

x

-intercepts

d

in terms of the constants

b

and

c

, we set up the equation

d = \frac{2}{3}k

. This simplifies to

b^2 + 4c = 36

. Using the point

(1, 8)

, we establish

c = 9 - b

. Substituting this into the first equation yields

b^2 - 4b = 0

, giving

b = 4

as the only positive solution.

Adım Adım Çözüm

Find the maximum value

k

of the function

f(x) = -x^2 + bx + c

in terms of

b

and

c

k = \frac{b^2}{4} + c

The maximum value of a quadratic function with a negative leading coefficient occurs at its vertex, where

x = -\frac{b}{2a} = \frac{b}{2}

Find the distance

d

between the

x

-intercepts of the graph in terms of

b

and

c

d = \sqrt{b^2 + 4c}

The

x

-intercepts are the roots of

-x^2 + bx + c = 0

, which are

x = \frac{b \pm \sqrt{b^2 + 4c}}{2}

. The distance between them is the difference of these roots.

Use the relation

d = \frac{2}{3}k

to find the value of the expression

b^2 + 4c

b^2 + 4c = 36

Substituting the expressions for

d

and

k

gives

\sqrt{b^2 + 4c} = \frac{1}{6}(b^2 + 4c)

. Solving this radical equation yields

b^2 + 4c = 36

Use the point

(1, 8)

to express

c

in terms of

b

c = 9 - b

Since the graph passes through

(1, 8)

, substituting

x = 1

and

y = 8

into

y = -x^2 + bx + c

yields

8 = -1 + b + c

, which simplifies to

c = 9 - b

Substitute

c = 9 - b

into

b^2 + 4c = 36

and solve for

b

b = 4

Substituting yields

b^2 + 4(9 - b) = 36 \implies b^2 - 4b = 0

. Solving for

b

gives

b = 0

b = 4

. Since

b

is a positive constant, we have

b = 4

Anahtar Kavram

Quadratic Functions and Graphs

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