In the xy-plane, the graph of a linear function f has a positive y-intercept and a positiv… — SAT

In the $xy$ -plane, the graph of a linear function $f$ has a positive $y$ -intercept and a positive $x$ -intercept. The area of the triangular region in the first quadrant bounded by the graph of $f$ and the coordinate axes is $36$ . If the graph of $f$ passes through the point $(4, 4)$ and has a slope less than $-1$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Answer: 12

Answer

The

y

-coordinate of the

y

-intercept of the graph of

f

12

The correct answer is

12

because the system of equations derived from the area constraint (

qr = 72

) and the point constraint (

q+r=18

) yields two possible values for the

y

-intercept:

6

12

. The condition that the slope must be less than

-1

means that the line must be steeper than a slope of

-1

, which requires the

y

-intercept to be larger than the

x

-intercept (

r > q

). Thus, the

y

-intercept is

12

Step-by-Step Solution

Express the equation of the line using intercept form.

\frac{x}{q} + \frac{y}{r} = 1

, where

q > 0

is the

x

-intercept and

r > 0

is the

y

-intercept.

Since the line intersects the positive axes, this form directly relates the intercepts to the coordinates of points on the line.

Use the area of the triangle to find a relationship between

q

and

r

qr = 72

The area of the right triangle formed by the axes and the intercepts is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}qr = 36

Substitute the point

(4, 4)

into the equation of the line and simplify using the area relationship.

q + r = 18

Plugging in

x = 4

and

y = 4

yields

\frac{4}{q} + \frac{4}{r} = 1 \implies 4(q + r) = qr

. Substituting

qr = 72

gives

4(q + r) = 72 \implies q + r = 18

Solve the system of equations

q + r = 18

and

qr = 72

(q, r) = (6, 12)

(q, r) = (12, 6)

Substituting

r = 18 - q

into

qr = 72

yields

q(18 - q) = 72 \implies q^2 - 18q + 72 = 0 \implies (q - 6)(q - 12) = 0

Apply the slope condition to determine the unique value of

r

r = 12

The slope of the line is

m = -\frac{r}{q}

. If

q = 12

and

r = 6

, then

m = -0.5

, which is not less than

-1

. If

q = 6

and

r = 12

, then

m = -2

, which is less than

-1

. Thus,

r = 12

is the correct

y

-intercept.

Key Concept

Using intercepts and area to determine the equation of a linear function under constraints.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic