Step-by-step explanation:
to answer the originated question first :
we know it is linear from the chart, because the function is a straight line.
we know it theoretically, when we look at x and y data pairs, and the change rate is the same constant for every x, y pair to each other.
the rate of change is also called "slope" or "incline". and it is described as ratio of "y coordinate change / x coordinate change".
it describes in fact how many units y changes, when x changes a certain amount of units (like 1).
having said all that, the most usual form as equation for such a line is the slope-intercept form.
y = ax + b
a is the slope of the line and always the factor of x.
b is the y-intercept (or the "initial value", as it is called here). it is simply the y value when x = 0.
we have 2 marked points on the line : (-2, 0) and (0, 3).
let's use them to find the slope.
going from the first to the second point
x changes by +2 (from -2 to 0).
y changes by +3 (from 0 to 3).
so, the slope (a) is +3/+2 = 3/2.
the initial value or y-intercept we get from the second point directly, as this is the point, where x = 0.
so, the initial value or y-intercept (b) is 3.
our line equation is
y = 3/2 x + 3 or 3x/2 + 3
now, we need to define a line with the same b but an a that is lower than 3/2.
since 3/2 is larger than 1, my choice for the new a would be 1 (lower than 3/2), because it is normally the easiest number to deal with.
so, the new line with lower change rate but the same initial value would be
y= x + 3
we could put it to some extreme and actually choose a = 0 (a change rate of 0).
our line equation would simply be
y = 3
this is a flat, horizontal line (parallel to the x-axis) that goes through y=3 on the y-axis.
as you can imagine, y never charged no matter what value we pick for x, so the change rate is 0. and clearly lower than 3/2.
your choice.
but how can you not know anything about this ? have you not been in the class ? but then you need to solve these questions ? that is not right.
I hope I could explain this now to you.
