The input to the function must reach the same original value after multiplication by b. Thus, a point formerly evaluated at x is represented in the new graph when the displayed input equals x divided by b. This reciprocal relationship explains why increasing b produces a narrower graph rather than moving points by b units.
A change inside the function, such as f(bx), alters the input coordinate associated with each output. A vertical transformation instead changes the function’s output values. For horizontal compression, corresponding points retain the same y-coordinate while their x-coordinates change, allowing the two types of scale changes to be identified from the equation and coordinate behavior.
The parameter b controls the scale of the horizontal coordinates. When b is greater than 1, every original x-coordinate is divided by b, so larger values produce greater narrowing toward the y-axis. Comparing different values of b therefore reveals how an equation’s parameter controls the degree of geometric change without altering the associated vertical values.
These transformations reverse one another through reciprocal scale factors. Replacing x with bx, for b greater than 1, reduces displayed horizontal distances by a factor of b. Applying the corresponding reciprocal factor restores the original horizontal spacing. This relationship helps students interpret paired equations and check whether a proposed transformation correctly reverses the first one.
First identify the coefficient multiplying the input inside the function. Next, take several known points from the original graph and divide each x-coordinate by that coefficient, leaving every y-coordinate unchanged. Plot the resulting points and connect them according to the original graph’s shape. This coordinate-based procedure links the algebraic substitution directly to the new geometry.
It provides a systematic way to connect changes in an equation with changes in a graph. By comparing the input coefficients, students can determine which function is narrower and quantify the difference using the reciprocal coordinate factor. The method supports analysis of function families, parameter effects, coordinate geometry, and models in which horizontal scale changes.