In the xy-plane, the graph of the quadratic function f(x) = ax^2 + bx + c, where a, b, and… — SAT

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has a vertex at $(2, -3)$ and passes through the point $(4, 5)$ . If the graph of a second quadratic function, $g$ , is obtained by translating the graph of $f$ horizontally by $3$ units to the right and vertically by $5$ units up, what is the value of $g(5)$ ?

A
-8
2Answer
C
8
D
74

Answer

The value of

g(5)

2

The vertex of the graph of

f

is at

(2, -3)

, which means

f(2) = -3

. The graph of

g

is obtained by translating the graph of

f

3

units to the right and

5

units up, so its equation is

g(x) = f(x - 3) + 5

. To find

g(5)

, we substitute

x = 5

into this relation, which gives

g(5) = f(5 - 3) + 5 = f(2) + 5

. Since

f(2) = -3

, we have

g(5) = -3 + 5 = 2

. Alternatively, translating the vertex of

f

(2, -3)

3

units to the right and

5

units up gives the vertex of

g

(2 + 3, -3 + 5) = (5, 2)

. Since the vertex of the parabola

g

occurs at

x = 5

, the value of

g(5)

is the

y

-coordinate of the vertex, which is

2

Step-by-Step Solution

Identify the vertex of the function

f

and write its vertex form equation.

The vertex of

f

(2, -3)

, so the vertex form of the function is

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

This allows us to find the specific equation of

f(x)

if needed, and also tells us that the value of

f(2)

-3

Set up the equation for the translated function

g(x)

A horizontal translation of

3

units to the right and a vertical translation of

5

units up is represented by

g(x) = f(x - 3) + 5

This defines the function

g

in terms of the function

f

using standard translation rules.

Evaluate

g(5)

using the relation from step 2.

g(5) = f(5 - 3) + 5 = f(2) + 5

This simplifies the calculation by using the value of

f

at the vertex.

Substitute the value of

f(2)

to find

g(5)

g(5) = -3 + 5 = 2

Since the vertex of

f

is at

(2, -3)

, we know

f(2) = -3

, which yields the final result.

Key Concept

Quadratic functions can be analyzed and transformed using their vertex form and function translation rules.

Estimated Time:1m 30s

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