For the quadratic function f, the table below shows three points that lie on its graph in … — SAT

For the quadratic function $f$ , the table below shows three points that lie on its graph in the $xy$ -plane, where $k$ is a constant.

$x$	$f(x)$
$2$	$0$
$6$	$0$
$1$	$-15$

If the vertex of the graph of $y = f(x)$ is $(4, k)$ , what is the value of $k$ ?

Cevap: 12

Cevap

The value of

k

12

The table indicates that the function has

x

-intercepts at

x = 2

and

x = 6

because

f(2) = 0

and

f(6) = 0

. Consequently, the quadratic function can be written in factored form as

f(x) = a(x - 2)(x - 6)

for some constant

a

. Using the point

(1, -15)

to find

a

, we substitute

x = 1

and

f(1) = -15

, which yields

-15 = a(1 - 2)(1 - 6)

, or

-15 = 5a

, so

a = -3

. Since the vertex of the graph is

(4, k)

, the value of

k

is the function value at the vertex's

x

-coordinate, which is

f(4)

. Evaluating the function gives

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

Adım Adım Çözüm

Write the quadratic function in factored form using the given

x

-intercepts.

f(x) = a(x - 2)(x - 6)

The table shows that

f(2) = 0

and

f(6) = 0

, meaning the graph has

x

-intercepts at

x = 2

and

x = 6

Substitute the point

(1, -15)

into the factored equation to find the value of the constant

a

a = -3

Substituting

x = 1

and

f(x) = -15

gives

-15 = a(1 - 2)(1 - 6) \implies -15 = 5a \implies a = -3

Calculate the value of

k

by finding the function value at the vertex

x = 4

k = 12

Since the vertex is

(4, k)

, the value of

k

f(4)

. Substituting

x = 4

into

f(x) = -3(x - 2)(x - 6)

yields

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

Anahtar Kavram

Using intercepts and an additional point to determine the equation of a quadratic function, and evaluating it at the vertex.

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