The function f is defined by f(x) = |x - 2| - 3. The function g is defined by g(x) = -f(x … — SAT

The function $f$ is defined by $f(x) = |x - 2| - 3$ . The function $g$ is defined by $g(x) = -f(x + 1) + 2$ . What is the maximum value of $g(x)$ ?

A
-1
B
1
C
3
5Cevap

Cevap

The vertex of the absolute value function

f(x) = |x - 2| - 3

is at

(2, -3)

, meaning its minimum value is

-3

. The transformed function

g(x) = -f(x + 1) + 2

shifts the graph 1 unit to the left, reflects it vertically across the

x

-axis (which turns the minimum value of

-3

into a maximum value of

3

), and translates it 2 units upward. This results in a final maximum value of

3 + 2 = 5

Adım Adım Çözüm

Find the minimum value of

f(x)

and the location of its vertex.

The vertex of

f(x) = |x - 2| - 3

is at

(2, -3)

, which means

f(x)

has a minimum value of

-3

x = 2

Since the coefficient of the absolute value expression is positive, the graph of

f

opens upward, making the vertex a minimum point.

Apply the horizontal translation to find the new vertex position.

The term

f(x + 1)

shifts the graph 1 unit to the left, moving the vertex from

x = 2

x = 1

Adding a constant inside the function argument,

f(x + h)

where

h > 0

, shifts the graph horizontally to the left by

h

units.

Apply the vertical reflection to the minimum value.

The term

-f(x + 1)

reflects the graph across the

x

-axis. The vertex point

(1, -3)

becomes

(1, 3)

, and the minimum value of

-3

becomes a maximum value of

3

Multiplying the function by

-1

reflects the graph vertically, swapping all positive and negative

y

-values and changing the minimum value to a maximum value.

Apply the vertical translation to find the final maximum value.

Adding 2 to the function,

g(x) = -f(x + 1) + 2

, shifts the entire graph vertically upward by 2 units. The vertex moves from

(1, 3)

(1, 5)

Adding a constant to the outside of the function shifts the graph vertically, so the maximum value increases from 3 to 5.

Anahtar Kavram

Function Notation and Transformations

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