The function f is defined by f(x) = |x - 6| + 3. In the xy-plane, the graph of the functio… — SAT

The function $f$ is defined by $f(x) = |x - 6| + 3$ . In the $xy$ -plane, the graph of the function $g$ is obtained by reflecting the graph of $f$ across the $x$ -axis and then translating it vertically upward by 11 units. If $g(a) = 0$ and $a > 0$ , what is the value of $a$ ?

Answer: 14

Answer

The value of

a

is 14.

Reflecting

f(x) = |x - 6| + 3

across the

x

-axis negates the entire expression, giving

-f(x) = -|x - 6| - 3

. Translating this graph upward by 11 units adds 11 to the expression, which yields

g(x) = -|x - 6| - 3 + 11 = -|x - 6| + 8

. Setting

g(a) = 0

results in

-|a - 6| + 8 = 0

, which simplifies to

|a - 6| = 8

. The solutions to this equation are

a = 14

and

a = -2

. Since the problem specifies that

a > 0

, the correct answer is 14.

Step-by-Step Solution

Reflect the function

f(x)

across the

x

-axis.

-f(x) = -|x - 6| - 3

Reflecting a graph across the

x

-axis negates the entire function expression, transforming

y = f(x)

y = -f(x)

Translate the reflected function upward by 11 units to obtain

g(x)

g(x) = -|x - 6| + 8

Translating a function vertically upward by

k

units adds

k

to the expression, so

g(x) = -f(x) + 11 = -|x - 6| - 3 + 11 = -|x - 6| + 8

Set

g(a) = 0

and solve for

a

|a - 6| = 8 \implies a = 14

a = -2

Solving the equation

-|a - 6| + 8 = 0

requires isolating the absolute value term to get

|a - 6| = 8

. This splits into two cases:

a - 6 = 8

and

a - 6 = -8

Apply the constraint

a > 0

to identify the final answer.

a = 14

The problem specifies that

a

must be positive, which excludes the solution

a = -2

and leaves

a = 14

Key Concept

Function transformations including reflections across the axes and vertical translations.

Estimated Time:1m 30s

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