The function f is defined by f(x) = x^2 - 4x + 7. The function g is defined by g(x) = f(x … — SAT

The function $f$ is defined by $f(x) = x^2 - 4x + 7$ . The function $g$ is defined by $g(x) = f(x - 3) + 2$ . If the minimum value of $f$ is $v$ , and the minimum value of $g$ occurs at $x = k$ , what is the value of $v + k$ ?

A
1
B
2
C
7
8Answer

Answer

To find the minimum value of

f(x) = x^2 - 4x + 7

, we can complete the square to write it in vertex form:

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

. The vertex of the graph of

f

(2, 3)

, meaning the minimum value of

f

v = 3

, occurring at

x = 2

. The function

g

is defined as

g(x) = f(x - 3) + 2

, which represents a shift of the graph of

f

to the right by 3 units and up by 2 units. Since the minimum of

f

occurs at

x = 2

, the minimum of

g

occurs at

x = 2 + 3 = 5

, so

k = 5

. Thus, the value of

v + k

3 + 5 = 8

Step-by-Step Solution

Find the vertex form of the quadratic function

f(x)

to identify its minimum value

v

and the x-coordinate where it occurs.

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

, so the minimum value is

v = 3

, which occurs at

x = 2

Completing the square allows us to read the vertex

(h, k)

of the parabola directly, where

h

is the x-coordinate of the vertex and

k

is the minimum value.

Determine the x-coordinate

k

where the minimum value of

g(x)

occurs using function transformations.

k = 2 + 3 = 5

The function

g(x) = f(x - 3) + 2

shifts the graph of

f

to the right by 3 units. Therefore, the minimum point is translated from

x = 2

x = 2 + 3 = 5

Calculate the sum

v + k

3 + 5 = 8

Adding the minimum value of

f

(

v = 3

) to the x-coordinate of the minimum of

g

(

k = 5

) gives the final required value.

Key Concept

Identifying the vertex and minimum values of quadratic functions, and applying horizontal and vertical translations to their graphs.

Estimated Time:1m 30s

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