The function f is defined for all real numbers, and the graph of y = f(x) in the xy-plane … — SAT

The function $f$ is defined for all real numbers, and the graph of $y = f(x)$ in the $xy$ -plane has a single minimum at the point $(5, -2)$ . The function $g$ is defined by $g(x) = -3f(2x - 4) + 7$ . What is the $y$ -coordinate of the maximum point on the graph of $y = g(x)$ ?

Cevap: 13

Cevap

The correct answer is 13.

The graph of

y = f(x)

has a minimum at

(5, -2)

, which means

f(5) = -2

and

f(x) \ge -2

for all

x

. The function

g(x) = -3f(2x-4) + 7

includes a vertical stretch by a factor of

3

, a vertical reflection across the

x

-axis, and a vertical shift upward by

7

units. Because of the vertical reflection, the minimum value of the original function becomes the maximum value of the transformed function. Applying the vertical transformations to the

y

-coordinate of the minimum point yields

-3(-2) + 7 = 6 + 7 = 13

Adım Adım Çözüm

Identify the minimum point and minimum value of the original function

f(x)

f(5) = -2

, and

f(x) \ge -2

for all real numbers

x

The problem states that the graph of

y = f(x)

has a single minimum at the point

(5, -2)

Determine the transformed

x

-coordinate corresponding to the original input of

5

2x - 4 = 5 \implies 2x = 9 \implies x = 4.5

Setting the argument of the function

f(2x-4)

equal to the original minimum input of

5

allows us to find the corresponding input

x

for the function

g

Apply the vertical transformations to find the output value of

g(x)

x = 4.5

g(4.5) = -3f(5) + 7 = -3(-2) + 7 = 6 + 7 = 13

Substituting

f(5) = -2

into the definition of

g(x)

gives the vertical transformation of the point.

Confirm that the point is indeed the maximum of the transformed function

g(x)

Since

f(2x-4) \ge -2

, multiplying by

-3

yields

-3f(2x-4) \le 6

. Adding

7

yields

g(x) \le 13

, confirming that

13

is the maximum value.

Multiplying a function by a negative number reflects its graph vertically, changing a minimum point into a maximum point.

Anahtar Kavram

Applying horizontal and vertical transformations to function coordinates, and understanding how vertical reflections affect the extrema (minima and maxima) of a graph.

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