What is expression of linear function if its graph is passing through the points (-1,3) and (3,1)?
The equation of the line passing through the points (x1, y1) and (x2, y2) is given by ( y - y1) = [( y2 - y1)/(x2 - x1)]( x - x1)
As the line passes through ( -1 , 3) and (3 , 1) we can substitute the values we have to get :
( y - 3) = [( 1 - 3)/(3 + 1)]( x + 1)
=> 4( y - 3) = -2( x + 1)
=> 2y - 6 = -x - 1
=> 2y = -x + 5
=> y = -x/2 + 5/2
=> f(x) = -x/2 + 5/2
The required function is f(x)= -x/2 + 5/2
The standard form of a linear function f(x) is:
f(x) = ax + b
y = mx + n, where m represents the slope of the line and n represents the y intercept.
In this case, the graph of the function is passing through the given points.
By definition, a point belongs to a curve if the coordinates of the point verify the equation of the curve.
(-1,3) is on the line y = ax+b if and only if:
3 = a*(-1) + b
-a + b = 3 (1)
(3,1) belongs to the line y = ax+b if and only if:
1 = a*3 + b
3a + b = 1 (2)
We'll multiply (1) by 3:
-3a + 3b = 9 (3)
We'll add (3) to (2):
3a + b - 3a + 3b = 1 + 9
We'll eliminate like terms:
4b = 10
b = 5/2
From (1)=>a = b - 3
a = 5/2 - 3
a = -1/2
The function f(x) whose graph is passing through the given points is:
f(x) = -(1/2)*x + 5/2