Question

When analyzing three linear functions, Michele noticed that they all intersected at the point (5, 3). What conclusion can Michele come to about the linear function with the smallest slope?
A. It has the middle y-intercept.

B. It has the highest y-intercept.

C. It has the lowest y-intercept.

D. Michele cannot conclude anything.

Answer 1

The slope of the line is the ratio of the change in the values of y to the change in the values of x. Since the lines intersect at a common point (5, 3), the line with the lowest or the smallest slope has the highest value for the y - intercept. Thus, the answer is letter B. 

Answer 2

Answer: option B. it has the highest y-intercept.

Explanation:

1) point -slope equation of the line

y - y₁ = m (x - x₁)

2) Replace (x₁, y₁) with the point (5,3):

y - 5 = m (x - 3)

3) Expand using distributive property and simplify:

y - 5 = mx - 3m ⇒ y = mx + 5 - 3m

4) Compare with the slope-intercept equation of the line: y = mx + b, where m is the slope and b is the y-intercept

⇒ slope = m

⇒ b = 5 - 3m = y - intercept.

Therefore, for the same point (5,3), the greater m (the slope of the line) the less b (the y-intercept); and the smaller m (the slope) the greater the y - intercept.

Then, the conclusion is: the linear function with the smallest slope has the highest y-intercept (option B).