How do you write a linear function f with the values #f(2)=-1# and #f(5)=4#?

Question

Kennedy Adams · Accepted Answer

Use the two points to compute the slope, m, then use one of the points in the form #y = m(x) + b# to find the value of b.

The equation for the slope, m, of a line is: #m = (y_1 - y_0)/(x_1 - x_0)" [1]"# The equation #f(2) = -1# tells us that #x_0 = 2 and y_0 = -1#; substitute this into equation [1]: #m = (y_1 - -1)/(x_1 - 2)" [2]"# The equation #f(5) = 4# tells us that #x_1 = 5 and y_1 = 4#; substitute this into equation [2]: #m = (4 - -1)/(5 - 2)" [3]"# #m = 5/3# Substitute #5/3# for m into the equation #y = m(x) + b# #y = 5/3x + b" [4]"# Substitute 2 for x and -1 for y and the solve for b: #-1 = 5/3(2) + b# #b = -13/3# Substitute #-13/3# for b in equation [4]: #y = 5/3x -13/3" [5]"# Check: #-1 = 5/3(2) -13/3# #4 = 5/3(5) - 13/3# #-1 = -1# #4 = 4# This checks Equation [5] is the answer.

Abigail Allen · Answer

undefined To write a linear function $ f $ with the given values $ f(2) = -1 $ and $ f(5) = 4 $, you can first find the slope $ m $ using the formula $ m = \frac{{f(5) - f(2)}}{{5 - 2}} $, then use the point-slope form of a linear equation $ y - y_1 = m(x - x_1) $ with one of the points $ (2, -1) $ or $ (5, 4) $.