Consider a function f (x) which can be defined by two points A (4; 1) and B (0; 3). We can say that …
f (x) is a linear function because it can be defined by two points.
To determine the equation of the function f(x) defined by two points A(4, 1) and B(0, 3), we can use the point-slope form of a linear equation.
Step 1: Find the slope (m) of the line passing through points A and B using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates A(4, 1) and B(0, 3):
m = (3 - 1) / (0 - 4)
m = 2 / -4
m = -1/2
Step 2: Substitute the slope (m) and one of the given points (A or B) into the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
Using point A (4, 1):
y - 1 = -(1/2)(x - 4)
Step 3: Simplify the equation:
y - 1 = -(1/2)x + 2
Step 4: Rewrite the equation in slope-intercept form (y = mx + b) by isolating y:
y = -(1/2)x + 2 + 1
y = -(1/2)x + 3
Therefore, the equation of the function f(x) defined by the two points A(4, 1) and B(0, 3) is:
f(x) = -(1/2)x + 3