APPM 1350: Graph Transformations, Trigonometry, Limits & Continuity

What does y = f(x) + k do to the graph?

It shifts the graph up k units.

What does y = f(x) - k do to the graph?

It shifts the graph down k units.

What is the relationship between sin θ and csc θ?

csc θ = 1/sin θ.

What does y = f(x - k) do to the graph?

It shifts the graph right k units.

What does y = f(x + k) do to the graph?

It shifts the graph left k units.

What does y = cf(x) do to the graph when c > 1?

It stretches the graph vertically.

What does y = (1/c)f(x) do to the graph?

It compresses the graph vertically.

What does y = f(cx) do to the graph?

It compresses the graph horizontally.

What does y = f(x/c) do to the graph?

It stretches the graph horizontally.

What is the definition of an even function?

A function f is even if f(−x) = f(x) for all x, making its graph symmetric about the y-axis.

What does y = -f(x) do to the graph?

It reflects the graph about the x-axis.

What does y = f(-x) do to the graph?

It reflects the graph about the y-axis.

What does y = |f(x)| do to the graph?

It reflects the negative function values about the x-axis.

What is the informal definition of a limit?

lim x→a f(x) = L if the values of f(x) approach L as x approaches a.

What is the precise definition of a limit?

If for every ϵ > 0 there is a corresponding δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ϵ, then lim x→a f(x) = L.

What does the Squeeze Theorem state?

If f(x) ≤ g(x) ≤ h(x) when x is near a and lim x→a f(x) = lim x→a h(x) = L, then lim x→a g(x) = L.

What is the theorem regarding sin θ as θ approaches 0?

lim θ→0 (sin θ/θ) = 1.

What is the definition of continuity at a point x = a?

A function f is continuous at x = a if f(a) = lim x→a f(x).

What are the three parts of the definition of continuity?

1. lim x→a f(x) exists. 2. f(a) exists. 3. f(a) = lim x→a f(x).