The function f is defined by f(x) = x^2 - 6x + c, where c is a constant. In the xy-plane, … — SAT

The function $f$ is defined by $f(x) = x^2 - 6x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $f$ has vertex $A$ . The function $g$ is defined by $g(x) = -f(x - 6)$ , and its graph has vertex $B$ . If the distance between points $A$ and $B$ is $10$ , and $c > 10$ , what is the value of $c$ ?

Cevap: 13

Cevap

The value of

c

13

Completing the square for

f(x) = x^2 - 6x + c

gives

f(x) = (x - 3)^2 + c - 9

, which shows that vertex

A

is located at

(3, c - 9)

. The transformation

g(x) = -f(x - 6)

translates the graph

6

units to the right and reflects it vertically, giving vertex

B

the coordinates

(3 + 6, -(c - 9)) = (9, 9 - c)

. Using the distance formula, the distance between

A

and

B

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = \sqrt{36 + (18 - 2c)^2}

. Setting this distance equal to

10

and squaring both sides gives

36 + (18 - 2c)^2 = 100

, which simplifies to

(18 - 2c)^2 = 64

. Taking the square root of both sides gives

18 - 2c = 8

18 - 2c = -8

, yielding solutions of

c = 5

c = 13

. Since the question specifies that

c > 10

, the correct value must be

13

Adım Adım Çözüm

Rewrite the function

f(x) = x^2 - 6x + c

in vertex form by completing the square.

f(x) = (x - 3)^2 + c - 9

, which gives the coordinates of vertex

A

(3, c - 9)

Completing the square reveals the vertex

(h, k)

of a quadratic function in the form

y = a(x - h)^2 + k

Determine the vertex

B

of the graph of

g(x) = -f(x - 6)

by applying transformations to vertex

A(3, c - 9)

The horizontal shift of

f(x - 6)

moves the vertex to

(3 + 6, c - 9) = (9, c - 9)

. The reflection of

-f(x - 6)

negates the

y

-coordinate of the vertex, resulting in

B(9, 9 - c)

The transformation

f(x - h)

shifts a graph right by

h

units, and the transformation

-f(x)

reflects it across the

x

-axis.

Use the distance formula to set up an equation for the distance between

A(3, c - 9)

and

B(9, 9 - c)

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = 10 \implies \sqrt{6^2 + (18 - 2c)^2} = 10

The distance

d

between points

(x_1, y_1)

and

(x_2, y_2)

is given by

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Solve the radical equation

\sqrt{36 + (18 - 2c)^2} = 10

for

c

36 + (18 - 2c)^2 = 100 \implies (18 - 2c)^2 = 64 \implies 18 - 2c = 8

18 - 2c = -8

. This yields

c = 5

c = 13

Squaring both sides eliminates the square root, allowing us to solve the resulting quadratic equation.

Apply the constraint

c > 10

to choose the correct value for

c

c = 13

The question specifies that

c

must be greater than

10

, which excludes

c = 5

Anahtar Kavram

Using vertex form of quadratic equations to determine vertex coordinates and applying transformations (horizontal shifts and vertical reflections) to find key graphical points.

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