Linear FunctionWhat is the linear function f if the graph of f passes through the point (1;3) and (2;1) ?
The basic equation for a line is y = mx+b where m is the slope and b is the y-intercept. First let's calculate the slope:
m = (y2-y1)/(x2-x1) = (3-1)/(1-2) = 2/-1 = -2
We can input the slope into the general equation to find the y-intercept:
y = mx + b = -2x + b
Input the first set of points (1,3):
3 = -2(1) + b
3 = -2 + b
b = 5
The equation of the linear function passing through both points given is: y = -2x + 5.
A linear function is determined when it's coefficients are determined.
y = f(x) = mx + n
So, in order to determine y, we'll have to calculate the coefficients m and n.
Since the function is determined by the points (1,3) and (2,1), that means that if we'll substitute the coordinates of the points into the expression of the function, we'll get the relations:
f(1) = 3
f(1) = m*1 + n
m + n = 3 (1)
f(2) = 1
f(2) = 2m + n
2m + n = 1 (2)
We'll subtract (1) from (2):
2m + n -m - n = 1 - 3
We'll combine and eliminate like terms:
m = -2
We'll substitute m in (1):
m + n = 3
-2 + n = 3
n = 3+2
n = 5
The expression of the linear function is:
f(x) = -2x + 5