Scaling Linear Functions: Multiplying Inputs and Outputs
Scaling linear functions by multiplying inputs and outputs is a Grade 11 Algebra 1 topic from enVision Chapter 3. Vertical scaling (multiplying the output) gives y = a(mx + b) = amx + ab, multiplying both slope and y-intercept by a. Horizontal scaling (multiplying the input by c) gives y = m(cx) + b = mcx + b, changing only the slope. Starting from y = 2x + 3, vertical scaling by 3 produces y = 6x + 9 — slope triples and y-intercept triples. Horizontal scaling by 1/2 (doubling the input) turns y = x - 1 into y = 2x - 1 — slope doubles, y-intercept unchanged.
Key Concepts
For a linear function $y = mx + b$, scaling transformations affect the function differently depending on whether you multiply the input or output:.
Vertical scaling : $y = a(mx + b) = amx + ab$ stretches/compresses vertically by factor $|a|$ Horizontal scaling : $y = m(cx) + b = mcx + b$ stretches/compresses horizontally by factor $\frac{1}{|c|}$.
Common Questions
What is vertical scaling of a linear function?
Multiplying the entire function output by a factor a: y = a(mx + b) = amx + ab. Both the slope and y-intercept are multiplied by a.
What is horizontal scaling of a linear function?
Replacing x with cx: y = m(cx) + b = mcx + b. Only the slope changes (multiplied by c); the y-intercept stays the same.
Starting from y = 2x + 3, what is the result of vertical scaling by 3?
y = 3(2x + 3) = 6x + 9. Slope triples from 2 to 6, and y-intercept triples from 3 to 9.
How does replacing x with 2x in y = x - 1 transform the function?
y = 2x - 1. The slope doubles from 1 to 2, but the y-intercept remains -1. This is horizontal compression by factor 1/2.
Why does vertical scaling change the y-intercept but horizontal scaling does not?
Vertical scaling multiplies every y-value including y = b at x = 0. Horizontal scaling only affects the slope term mx, leaving the constant b unchanged.
What happens when a < 1 in vertical scaling?
The function is vertically compressed. Slope and y-intercept both decrease proportionally, making the line less steep.