To find the equation of an affine function from two distinct points A(x₁, y₁) and B(x₂, y₂), first calculate a = (y₂ − y₁)/(x₂ − x₁), then use either point to obtain b = y₁ − ax₁. The required equation is f(x) = ax + b. This method requires x₁ and x₂ to differ; when the x-coordinates are equal and the y-coordinates differ, the line is vertical and does not represent a function of x.
What do the coefficients a and b mean?
In f(x) = ax + b, the coefficient a measures the change in y for each unit change in x and determines the slope of the line. If a is positive, the line rises from left to right; if negative, it falls; if zero, it is horizontal. The coefficient b is f(0), the height at which the line crosses the y-axis, at (0, b). When a = 0, the expression describes a constant function, not a first-degree polynomial function.
Step by step: find the equation from two points
- Identify the coordinates A(x₁, y₁) and B(x₂, y₂), keeping each x paired with its corresponding y.
- Check that x₂ − x₁ is not zero. If it is zero, consider the vertical-line case before attempting division.
- Calculate the slope: a = (y₂ − y₁)/(x₂ − x₁). Use the same subtraction order in numerator and denominator.
- Substitute either point into y = ax + b and isolate b = y₁ − ax₁.
- Write f(x) = ax + b and substitute both original x-values to check that the resulting y-values match the given ones.
Example 1: two points with integer coordinates
Take A(1, 2) and B(4, 5). The change in y is 5 − 2 = 3, and the change in x is 4 − 1 = 3. Therefore, a = 3/3 = 1. Using A gives 2 = 1 · 1 + b, so b = 1. The equation is f(x) = x + 1.
Check by substituting both points: f(1) = 1 + 1 = 2 and f(4) = 4 + 1 = 5. On the graph, plot A and B, also plot (0, 1), the y-intercept, and draw the unique line through the two distinct points. Moving three units along x produces a rise of three units in y.
Example 2: points with fractional coordinates
Now use A(0, 1/2) and B(2, 3/2). The calculation is a = [(3/2) − (1/2)]/(2 − 0) = 1/2. Because the first point already has x = 0, its y-value immediately gives b = 1/2. Thus, f(x) = (1/2)x + 1/2.
Check: f(0) = 1/2 and f(2) = 1 + 1/2 = 3/2. On the coordinate plane, the line crosses the y-axis at 1/2 and rises one unit in y for every two units added to x. Fractions do not change the method: simply preserve the arithmetic and the graph scales.
Example 3: when the points do not define a function of x
Consider A(2, 1) and B(2, 4). The denominator of the slope formula would be 2 − 2 = 0, and division by zero is undefined. The points determine the vertical line x = 2, but not an equation of the form f(x) = ax + b: the same x = 2 would have both y = 1 and y = 4. On the graph, the points line up vertically, showing that the relation also fails the vertical line test.
There is an important qualification: if the two ordered pairs were exactly the same, they would not be two distinct points determining a unique line. In that case, the supplied information likewise cannot establish a unique equation.
How to check the answer on the graph and by substitution
- Find the intercept (0, b) and plot it on the y-axis.
- Use a = Δy/Δx to identify the vertical and horizontal changes; for example, a = 1/2 means rising 1 while moving 2 to the right.
- Plot the two original points, respecting each axis scale. Both points and (0, b) must lie on the same line.
- Compute f(x₁) and f(x₂); the results must be exactly y₁ and y₂. If either check fails, revisit signs, fractions, and coordinate pairings.
Common mistakes that change the result
Do not reverse only the y-subtraction when finding a: if you change y₂ − y₁ to y₁ − y₂, you must also change x₂ − x₁ to x₁ − x₂. Do not mistake b for the y-value at any arbitrary point: b is the value at x = 0. Do not reject a horizontal line, because y = b is a constant function. And do not assign a finite slope to a vertical line.
For a reference on equation form, the meaning of slope, and vertical lines, see OpenStax's Linear Functions section. The three numerical examples above were developed for this exercise.
