Question

Which function has the greatest rate of change? A. x y 3 3 4 6 5 9 B. y = 4 + 6x C. 12x + 6y = 18 D. A linear function that goes through point 1, negative 1 and point 0, 4

Answer 1

B. y = 4 + 6x

Explanation:

We know that the rate of change of a straight line is equal to the slope of the line.

Now, the general form of a straight line is y = mx + b, where m is the slope and b is the y-intercept.

Also, the slope of a line joined by
`( x_(1),y_(1) )` and
`( x_(2),y_(2) )` is given by
`m=(y_(2)-y_(1))/(x_(2)-x_(1))`.

So, using these, we will find the slopes of the lines given in the options.

A. Here, we are given two points (3,3) and (4,6). Then, the slope is given by,

`m_(1)=(6-3)/(4-3)` i.e.
`m_(1)=3`.

B. We have the equation of line y = 4 + 6x. On comparing it with the general form, we get that the slope is 6 i.e.
`m_(2)=6`.

C. We are given the standard form of a line. First, we convert it into the slope-intercept form ( or the general form ).

i.e. 12x + 6y = 18 → 2x + y = 3 → y = -2x + 3

Now, on comparing this equation with the general form of a line, we get that the slope is -2. i.e.
`m_(3)=-2`.

D. Here, we are given a linear function passing through (1,-1) and (0,4). Then the slope is given by,

`m_(1)=(4+1)/(0-1)` i.e.
`m_(4)=-5`.

Hence, we obtain that the function y=4+6x has the greatest slope or the greatest rate of change i.e.
`m_(2)=6`.