Question

Compare the rates of change of the following 2 items? Item 1:`y=-2x+6 color(white)("d")` Item 2: A line which passes through the points (2,2) and (1,6)

Which of the following is correct:
`color(white)("d")` A. The items decrease at the same rate. `color(white)("d")` B. Item 1 decreases faster.`color(white)("d")` C. Item 2 decreases faster.

Answer 1

Option C is true.

Explanation:

To enable direct comparison we need to convert the given 2 points into the gradient of an equation.

Standard form `y=mx+c` where `m` is the gradient.

Gradient is `("change in y")/("change in x") -> (y_2-y_1)/(x_2-x_1)`

Note that `x_1` is the least `x` value. So we travel from `x=1" to "x=2`

Thus `(x_1,y_1)=(1,6) and (x_2,y_2)=(2,2)` giving:

`m=(y_2-y_1)/(x_2-x_1) =(2-6)/(2-1) = (-4)/1`

The slope of the given equation is `-2 larr`Item 1
The slope between the given points is `-4larr` Item 2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Options given:

A: Items decrease at the same rate `color(red)(larr" False")`
B: Item 1 decreases faster `color(white)("ddddd.d")color(red)(larr" False")`
C: Item 2 decreases faster `color(white)("ddddddd")color(green)(larr" True")`