A linear function f(x) = mx + b satisfies the inequality f(x + 3) - f(x) \geq 6 for all re… — ACT

A linear function $f(x) = mx + b$ satisfies the inequality $f(x + 3) - f(x) \geq 6$ for all real numbers $x$ , and its graph passes through the point $(1, -2)$ . If the region bounded by the graph of $f$ , the line $x = -2$ , the $x$ -axis, and the $y$ -axis has an area of $20$ square units, what is the value of $b$ ?

A
-10
B
-8
-6Answer
D
-4
E
4

Answer

The value of

b

-6

To find the value of

b

, we first determine the constraints on the slope

m

. Since

f(x) = mx + b

, the difference

f(x+3) - f(x) = 3m

. Given

3m \geq 6

, we find

m \geq 2

. The graph of

f

passes through

(1, -2)

, which gives

-2 = m + b

, or

b = -2 - m

. Since

m \geq 2

, the

y

-intercept

b

must be less than or equal to

-4

, meaning the function is negative for the entire interval

[-2, 0]

. The region bounded by the graph of

f

x = -2

, the

x

-axis, and the

y

-axis forms a trapezoid below the

x

-axis. The vertical parallel sides of this trapezoid have lengths equal to the absolute values of the

y

-coordinates at

x = -2

and

x = 0

, which are

2m - b

and

-b

, respectively. The width is

2

. The area of the trapezoid is

\frac{(2m - b) + (-b)}{2} \times 2 = 2m - 2b

. Setting the area to

20

square units gives

2m - 2b = 20

, which simplifies to

m - b = 10

. Solving the system of equations

m + b = -2

and

m - b = 10

yields

m = 4

and

b = -6

Step-by-Step Solution

Find the constraint on the slope

m

using the given inequality

f(x+3) - f(x) \geq 6

3m \geq 6 \implies m \geq 2

The difference in function values over an interval of

3

for a linear function is

3

times the slope

m

Substitute the point

(1, -2)

into the equation

f(x) = mx + b

to find a relationship between

m

and

b

-2 = m(1) + b \implies b = -2 - m

The graph of the function must pass through the given coordinate point.

Determine the shape and boundaries of the region bounded by the graph of

f

x = -2

, the

x

-axis, and the

y

-axis.

A trapezoid below the

x

-axis with parallel vertical sides of lengths

2m - b

(at

x = -2

) and

-b

(at

x = 0

), and a horizontal width of

2

Since

m \geq 2

, the

y

-intercept

b

is at most

-4

, so the function is strictly negative on the interval

[-2, 0]

Set up the area formula for the trapezoid and set it equal to

20

to find another relation between

m

and

b

\frac{(2m-b) + (-b)}{2} \times 2 = 20 \implies 2m - 2b = 20 \implies m - b = 10

The area of a trapezoid is the average of the parallel side lengths multiplied by the width.

Solve the system of equations:

m + b = -2

and

m - b = 10

Adding the equations gives

2m = 8 \implies m = 4

. Substituting

m = 4

gives

b = -6

To find the specific value of

b

that satisfies both the point condition and the area condition.

Key Concept

Graphing linear equations, calculating area of bounded regions on the coordinate plane, and using linear slope and point-intercept forms.

Alternative Method

Instead of solving the system of equations algebraically, we can express the line in point-slope form as

y + 2 = m(x - 1)

. At

x = 0

y = -m - 2

, and at

x = -2

y = -3m - 2

. The heights of the trapezoid are

m + 2

and

3m + 2

(since they are below the

x

-axis and

m \geq 2

). The area of the trapezoid is

\frac{(m+2) + (3m+2)}{2} \times 2 = 4m + 4

. Setting

4m + 4 = 20

yields

4m = 16 \implies m = 4

, which gives

b = -m - 2 = -6

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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