The function f is defined by f(x) = 3^x - 4. In the xy-plane, the graph of the function g … — SAT

The function $f$ is defined by $f(x) = 3^x - 4$ . In the $xy$ -plane, the graph of the function $g$ is obtained by first reflecting the graph of $f$ across the $x$ -axis, then translating the graph vertically up by 10 units, and finally translating the graph horizontally to the right by 2 units. If $g(c) = 5$ , what is the value of $c$ ?

Cevap: 4

Cevap

The value of c is 4.

Reflecting the function

f(x) = 3^x - 4

across the

x

-axis changes its sign to

-f(x) = -3^x + 4

. Translating this graph vertically up by 10 units adds 10 to the function, yielding

-3^x + 14

. Finally, translating horizontally to the right by 2 units replaces

x

with

x - 2

, producing the function

g(x) = -3^{x-2} + 14

. Setting

g(c) = 5

gives the equation

-3^{c-2} + 14 = 5

. Subtracting 14 from both sides results in

-3^{c-2} = -9

, which simplifies to

3^{c-2} = 9

. Since

9 = 3^2

, the exponent

c-2

must equal 2, which gives

c = 4

Adım Adım Çözüm

Reflect the function

f(x) = 3^x - 4

across the

x

-axis

-f(x) = -(3^x - 4) = -3^x + 4

A reflection across the

x

-axis replaces

y

with

-y

, meaning the entire function is multiplied by

-1

Translate the reflected function vertically up by 10 units

-3^x + 4 + 10 = -3^x + 14

A vertical translation upward by

k

units adds

k

directly to the function expression.

Translate the function horizontally to the right by 2 units

g(x) = -3^{x-2} + 14

A horizontal translation to the right by

h

units replaces

x

with

x-h

in the function expression.

Set

g(c) = 5

and solve the exponential equation for

c

-3^{c-2} + 14 = 5 \implies 3^{c-2} = 9 \implies c - 2 = 2 \implies c = 4

Substitute

c

into

g(x)

, set the output to 5, isolate the exponential term, and equate exponents to find

c

Anahtar Kavram

Applying sequential function transformations (reflections, vertical translations, horizontal translations) algebraically and solving exponential equations.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru