In the xy-plane, the graph of the linear function f is perpendicular to the line with equa… — SAT

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $y = 2x - 7$ . If the region in the first quadrant bounded by the graph of $f$ , the $x$ -axis, and the $y$ -axis has an area of $36$ , what is the $x$ -intercept of the graph of $f$ ?

Cevap: 12

Cevap

The

x

-intercept of the graph of

f

12

The correct answer is

12

. Since the graph of

f

is perpendicular to the line

y = 2x - 7

, its slope is

-\frac{1}{2}

. The equation of the line is

f(x) = -\frac{1}{2}x + b

, which has a

y

-intercept of

(0, b)

and an

x

-intercept of

(2b, 0)

. In the first quadrant, these intercepts form a right triangle with the axes, having legs of length

b

and

2b

. The area of this triangle is

\frac{1}{2}(2b)(b) = b^2

. Setting the area equal to

36

gives

b^2 = 36

, so

b = 6

(since

b > 0

). The

x

-intercept is

2b = 2(6) = 12

Adım Adım Çözüm

Find the slope of the perpendicular line

f

The slope of

f

-\frac{1}{2}

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The given line has a slope of

2

, so the slope of

f

must be

-\frac{1}{2}

Express the

y

-intercept and

x

-intercept of

f

in terms of a single variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

, where

b > 0

The equation of

f

can be written as

f(x) = -\frac{1}{2}x + b

. Setting

x = 0

gives the

y

-intercept

b

. Setting

f(x) = 0

and solving for

x

gives the

x

-intercept

2b

Set up the area equation for the triangle in the first quadrant.

The area is represented by

b^2 = 36

The region bounded by the graph of

f

and the coordinate axes in the first quadrant forms a right triangle with perpendicular sides of lengths

b

and

2b

. The area of this triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2b \times b = b^2

Solve for

b

and calculate the

x

-intercept.

The

x

-intercept is

12

Solving

b^2 = 36

with the condition

b > 0

yields

b = 6

. The

x

-intercept is

2b

, which equals

2(6) = 12

Anahtar Kavram

Using perpendicular slopes and intercepts of linear functions to analyze geometric areas in the coordinate plane.

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