A linear function contains the points shown in the table below: | x | y | | :--- | :--- | | k - 2 | 5 | | k + 1 | 14 | | 2k + 3 | 35 | What is the y-intercept of the line representing this function?

A linear function contains the points shown in the table below: | x | y | | :--- | :--- | … — ACT

A linear function contains the points shown in the table below:

$x$	$y$
$k - 2$	$5$
$k + 1$	$14$
$2k + 3$	$35$

What is the $y$ -intercept of the line representing this function?

Answer: -4

Answer

The

y

-intercept of the line representing this function is

-4

By using the first two coordinates, the constant slope of the linear function is determined to be

3

. Equating the slope between the second and third coordinates to

3

yields the parameter

k = 5

. Substituting this back into the coordinate expressions gives the points

(3, 5)

and

(6, 14)

. Using the point-slope form, the equation of the line is

y = 3x - 4

, meaning the

y

-intercept is

-4

Step-by-Step Solution

Calculate the slope of the line using the first two points:

(k-2, 5)

and

(k+1, 14)

m = \frac{14 - 5}{(k+1) - (k-2)} = \frac{9}{3} = 3

Since the function is linear, the rate of change (slope) remains constant between any two points.

Express the slope using the second and third points,

(k+1, 14)

and

(2k+3, 35)

, set it equal to

3

, and solve for

k

\frac{35 - 14}{(2k+3) - (k+1)} = 3 \implies \frac{21}{k+2} = 3 \implies 3(k+2) = 21 \implies k+2 = 7 \implies k = 5

The slope between the second and third points must also equal the constant slope of

3

Substitute

k = 5

back into the coordinates to determine the actual points on the line.

The points are

(3, 5)

(6, 14)

, and

(13, 35)

This provides concrete coordinates that can be used to write the equation of the line.

Use the point-slope form with the point

(3, 5)

and slope

m = 3

to write the equation of the line, then convert to slope-intercept form.

y - 5 = 3(x - 3) \implies y - 5 = 3x - 9 \implies y = 3x - 4

Converting to slope-intercept form (

y = mx + b

) directly gives the

y

-intercept as the constant term

b

Key Concept

Determining the equation and intercepts of a line using the constant slope property of linear functions.

Alternative Method

Once the slope

m = 3

and a point such as

(3, 5)

are established, substitute these values directly into the slope-intercept equation

y = mx + b

to solve for

b

. This gives

5 = 3(3) + b \implies 5 = 9 + b \implies b = -4

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic