The graph of the quadratic function f in the xy-plane has a vertex at (4, 7). If the funct… — SAT

The graph of the quadratic function $f$ in the $xy$ -plane has a vertex at $(4, 7)$ . If the function $g$ is defined by $g(x) = f(x) + 5$ , what is the $y$ -coordinate of the vertex of the graph of $y = g(x)$ ?

Answer: 12

Answer

The correct answer is 12.

The vertex of the graph of the quadratic function

f

(4, 7)

. The function

g(x) = f(x) + 5

adds

5

to every output value of

f(x)

, which shifts the entire graph vertically upward by

5

units. Consequently, the vertex of the graph of

g

is translated from

(4, 7)

(4, 7 + 5)

, which simplifies to

(4, 12)

. Therefore, the

y

-coordinate of the vertex of the graph of

y = g(x)

12

Step-by-Step Solution

Identify the y-coordinate of the vertex of the graph of the function f.

The y-coordinate of the vertex of the graph of f is 7.

The vertex of the graph of f is given as (4, 7).

Determine the vertical translation from f(x) to g(x).

The graph of g(x) is shifted upward by 5 units relative to the graph of f(x).

The term + 5 in the definition g(x) = f(x) + 5 increases all y-values of the function by 5.

Calculate the y-coordinate of the vertex of the graph of g.

The new y-coordinate is 7 + 5 = 12.

Applying the vertical shift of 5 units to the original y-coordinate of 7 gives the new vertex's y-coordinate.

Key Concept

Vertical translations of quadratic functions and their graphs

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic