A linear function includes the points (4,−3) and (−9,4) . What fraction in lowest terms represents the output value of this function for an input of zero?

Question

help son

Answer 1

The strategy is to find the equation of the line passing through those points, then substitute zero for x to determine the corresponding y-value.

The points: (4, -3) and (-9,4).

First thing you need to get is the slope of the line through those points (change in y over change in x, or rise over run).

m = [4 - (-3)] / [-9 - 4] = 7 / -13

OR

m = [-3 - 4] / [ 4 - (-9)] = -7/ 13 (same result)

Now it is time to pick one of the given points along with the just determined slope to write the equation of the line. There are two basic methods (use the point-slope formula, or use slope-intercept formula). Some texts also teach the two-point formula.

Let's use (4, -3).

Point-slope would look like this:

y - (-3) = (-7/13)(x -4)

y + 3 = (-7/13)x + 28/13

y = (-7/13)x+28/13 - 39/13

y = (-7/13) x - 11/13

At this juncture if you let x = 0:

y = (-7/13)(0) - 11/13

y = -11/13

Note that if you use point-slope to start, when x = 4, y = -3.

-3 = (-7/13)(4) + b

-3=-28/13 + b

b = -39/13 + 28/13

b = -11/13

So the equation is:

y = (-7/13)x - 11/13 (same equation, same result).

Here is a Desmos graph showing how you can do some of this in a slightly different way (by naming the points with variables).

desmos.com/calculator/gt0ptd9i3n