Which expression describes a vertical compression of the graph by a factor a, where 0 < a < 1?

Which expression describes a vertical compression of the graph by a factor a, where 0 < a < 1?

• A. g(x) = a f(x), a > 1
• B. g(x) = a f(x), 0 < a < 1
• C. g(x) = f(x)/a
• D. g(x) = - a f(x)

Answer: (B)

Vertical compression by a factor a, with 0 < a < 1, means you scale every y-value by multiplying by a. The x-values stay the same, so the graph gets shorter and sits closer to the x-axis. This is described by g(x) = a f(x) when 0 < a < 1. Each point (x, f(x)) moves to (x, a·f(x)), shrinking the height of the graph. Why the other forms don’t fit: multiplying by a with a > 1 would stretch the graph vertically, making it taller. Dividing by a (which is the same as multiplying by 1/a) would also stretch the graph since 1/a > 1 when 0 < a < 1. Including a negative sign would reflect the graph across the x-axis in addition to scaling, which isn’t just a compression.

Vertical compression by a factor a, with 0 < a < 1, means you scale every y-value by multiplying by a. The x-values stay the same, so the graph gets shorter and sits closer to the x-axis.

This is described by g(x) = a f(x) when 0 < a < 1. Each point (x, f(x)) moves to (x, a·f(x)), shrinking the height of the graph.

Why the other forms don’t fit: multiplying by a with a > 1 would stretch the graph vertically, making it taller. Dividing by a (which is the same as multiplying by 1/a) would also stretch the graph since 1/a > 1 when 0 < a < 1. Including a negative sign would reflect the graph across the x-axis in addition to scaling, which isn’t just a compression.