Function Graphs · FuncLab

Transformations

Starting from y = f(x), the graph of y = a·f(b(x − h)) + k comes from four moves:

h shifts left or right, k shifts up or down, a stretches vertically (and flips over the x-axis when a < 0), b compresses horizontally by 1/b.

The faint curve is the original; the dots show where a key point moves.

Exponentials & logarithms

y = aˣ and y = logₐx (a > 0, a ≠ 1) are inverse functions: their graphs mirror each other in the line y = x.

For a > 1 both increase; for 0 < a < 1 both decrease. The exponential always passes through (0, 1), the logarithm through (1, 0).

Drag the point on the curve to see its mirror image in y = x.

The unit circle

The terminal side of angle θ meets the unit circle at P(cos θ, sin θ): P's height is sin θ and its x-coordinate is cos θ.

Let θ turn and record P's height against θ: the circle unrolls into the sine curve y = sin x. One full turn is 2π, so the period of sine is 2π.

Sinusoids

In y = A sin(ωx + φ) + k: A sets the amplitude, ω the period T = 2π/ω, φ the initial phase, and k shifts the graph up or down.

From y = sin x: shift left by φ, then compress x by 1/ω. If you compress first, the shift is only φ/ω.

Five-point method: set ωx + φ to 0, π/2, π, 3π/2, 2π to plot one period's key points.

Secant to tangent

Take P(x₀, f(x₀)) and Q(x₀ + h, f(x₀ + h)) on the curve. The slope of secant PQ is the average rate of change (f(x₀ + h) − f(x₀)) / h.

As h → 0, Q slides towards P and the secant approaches the tangent. Its slope is the derivative f′(x₀).

Monotonicity & extrema

The top graph is f(x), the bottom one its derivative f′(x).

Where f′(x) > 0, f is increasing; where f′(x) < 0, it is decreasing.

Where f′ changes from + to − there is a local maximum; from − to + a local minimum. A point with f′(x₀) = 0 but no sign change (x³ at 0) is not an extremum.

Bisection method

If f is continuous on [a, b] and f(a)·f(b) < 0, then f has at least one zero in (a, b) (the intermediate value theorem).

Bisection: take the midpoint m, check the sign of f(m), and keep the half where the signs still differ. Each step halves the interval until it is short enough.

Drag the endpoints, or press Next step to bisect.

Riemann sums

To find the area under a curve, split [a, b] into n equal pieces and approximate each by a rectangle. The total is a Riemann sum.

As n grows the rectangles narrow and the sum approaches the area; the limit is the definite integral ∫ₐᵇ f(x) dx.

Compare taking the height at the left end, right end and midpoint.