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 vertex $(3, 12)$ and passes through the point $(1, 0)$ . The function $g$ is defined by $g(x) = f(x + d) - 4$ , where $d$ is a constant. If the $y$ -intercept of the graph of $g$ is $(0, 5)$ and the vertex of the graph of $g$ lies in the second quadrant, what is the value of $d$ ?

Cevap: 4

Cevap

The value of

d

is 4.

First, the equation of the function

f(x)

is determined in vertex form. Since the vertex of

f

(3, 12)

, we write

f(x) = a(x - 3)^2 + 12

. Substituting the point

(1, 0)

into this equation gives

0 = a(1 - 3)^2 + 12

, which yields

a = -3

. Thus,

f(x) = -3(x - 3)^2 + 12

. The transformation

g(x) = f(x + d) - 4

shifts the graph of

f

left by

d

units and down by 4 units, so the vertex of the graph of

g

(3 - d, 8)

. For this vertex to lie in the second quadrant, the

x

-coordinate must be negative, meaning

3 - d < 0

, or

d > 3

. The

y

-intercept of

g

(0, 5)

, so

g(0) = 5

. Since

g(0) = f(d) - 4

, we have

f(d) = 9

. Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

. Solving for

d

gives

d = 4

d = 2

. Since

d > 3

, the value of

d

must be 4.

Adım Adım Çözüm

Write the function

f(x)

in vertex form and substitute the point

(1, 0)

to solve for

a

f(x) = -3(x - 3)^2 + 12

The vertex

(h, k)

is given as

(3, 12)

, and the point

(1, 0)

lies on the graph.

Determine the vertex of

g(x)

based on the horizontal and vertical translations of

f(x)

The vertex of

g

(3 - d, 8)

The transformation

g(x) = f(x + d) - 4

shifts the vertex of

f(x)

left by

d

units and down by 4 units.

Establish the constraint on

d

using the quadrant of the vertex of

g

d > 3

For the vertex

(3 - d, 8)

to lie in the second quadrant, the

x

-coordinate must be negative.

Set up an equation for

d

using the

y

-intercept of

g(x)

f(d) = 9

The

y

-intercept is

(0, 5)

, so

g(0) = 5

. Substituting this into

g(x) = f(x + d) - 4

gives

5 = f(d) - 4

Solve

f(d) = 9

for

d

d = 2

d = 4

Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

Select the correct value of

d

that satisfies the quadrant constraint.

d = 4

Since

d > 3

, the value

d = 2

is discarded, leaving

d = 4

as the only valid solution.

Anahtar Kavram

Vertex form and transformations of quadratic functions

Soru

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