In the xy-plane, the graph of the linear function f(x) = px + q, where p and q are constan… — SAT

In the $xy$ -plane, the graph of the linear function $f(x) = px + q$ , where $p$ and $q$ are constants, passes through the point $(2, 7)$ . Line $L$ is parallel to the graph of $f$ and has a $y$ -intercept that is $3$ units below the $y$ -intercept of the graph of $f$ . If the $x$ -intercept of line $L$ is $(-6, 0)$ , what is the value of $p$ ?

Cevap: 0.5

Cevap

The value of

p

0.5

(or the fraction

1/2

The correct answer is

0.5

(or

1/2

). The graph of

f(x) = px + q

passes through

(2, 7)

, which means

7 = 2p + q

, or

q = 7 - 2p

. Line

L

is parallel to

f

, so its slope is

p

, and its

y

-intercept is

q - 3

. Thus, the equation of line

L

y = px + q - 3

. Since line

L

has an

x

-intercept at

(-6, 0)

, we can substitute

x = -6

and

y = 0

into its equation, yielding

0 = -6p + q - 3

. Substituting

q = 7 - 2p

into this equation gives

0 = -6p + (7 - 2p) - 3

, which simplifies to

-8p + 4 = 0

. Solving for

p

gives

8p = 4

, or

p = 0.5

Adım Adım Çözüm

Express the relationship between

p

and

q

using the given point

(2, 7)

on the graph of

f

q = 7 - 2p

The point

(2, 7)

must satisfy the equation

f(x) = px + q

Formulate the equation of line

L

using the parallel slope and the shifted

y

-intercept.

y = px + q - 3

Parallel lines have equal slopes, and the

y

-intercept of

L

3

units below the

y

-intercept of

f

, which is

q

Substitute the

x

-intercept

(-6, 0)

into the equation of line

L

-6p + q - 3 = 0

The

x

-intercept is a point on the line where

y = 0

Substitute the expression for

q

from Step 1 into the equation from Step 3 and solve for

p

p = 0.5

(or

\frac{1}{2}

)

Solving the linear equation

-6p + (7 - 2p) - 3 = 0

simplifies to

-8p + 4 = 0

, giving

p = 0.5

Anahtar Kavram

Understanding linear functions, their graphs, slopes of parallel lines, and intercepts.

Alternatif Yöntem

Instead of solving for

q

first, you can use the point-slope form. Line

L

passes through

(-6, 0)

and has slope

p

, so its equation is

y = p(x + 6)

, or

y = px + 6p

. The

y

-intercept of

L

6p

. The

y

-intercept of the graph of

f

q

. We are given that the

y

-intercept of

L

3

units below the

y

-intercept of

f

, so

6p = q - 3

. Since

f

passes through

(2, 7)

, we have

7 = 2p + q

, which means

q = 7 - 2p

. Substituting this into

6p = q - 3

gives

6p = (7 - 2p) - 3

, or

6p = 4 - 2p

. Adding

2p

to both sides gives

8p = 4

, which results in

p = 0.5

Tahmini Süre:2m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

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