Functions
Functions: 15 free interactive math resources. How independent and dependent variables relate — plot and morph graphs.
Guide
About this topic
Functions builds parameter↔graph intuition with linear, quadratic, and inverse tools before heavier algebra.
What you learn
- Slope and intercept for linear graphs
- Opening, vertex, and axis for quadratics
- Inverse proportion intuition
- From graph back to an equation
Subtopics
Key formulas
y = kx + by = ax² + bx + cy = k/x
FAQ
Linear before quadratic?
Usually yes — slope first, then parabolas.
Vertex form vs general form?
Switch tools to see the same parabola written two ways.
Good for projection?
Yes — ask what happens when a parameter flips sign.
Resources in this topic
15 free interactive resources in 3 groups
Linear functions
Quadratic functions
- Activity
Quadratic y = ax² + bx + c
Adjust a, b, c to see opening, axis of symmetry, and vertex.
Guide - Activity
Vertex form of a parabola
From y = a(x − h)² + k, see how translations move the parabola.
Guide - Activity
Maximum area under a parabola
P moves on a parabola; maximize area of △PAB (vertical height).
Open - Activity
Definition of a parabola
Equidistant from a focus and a directrix.
Open
Other functions
- Activity
Inverse variation y = k/x
Vary k to see the hyperbola branches and asymptotes.
Guide - Activity
Exponential function
y = aˣ: a > 1 increasing, 0 < a < 1 decreasing, always through (0, 1).
Open - Activity
Logarithmic functions
y = log_a x is the inverse of the exponential; it passes through (1,0).
Guide - Activity
Piecewise functions
Different formulas on each side of a breakpoint — see continuity and corners.
Guide - Activity
Function shifts and stretches
Using y=x², see how a, b, h, k stretch and shift the graph.
Guide - Activity
Sine function parameters
y = A sin(ωx + φ): see how amplitude, frequency, and phase change the graph.
Guide - Activity
Definition of an ellipse
Locus of points with PF₁ + PF₂ = 2a (constant).
Open - Activity
Exam functions: hyperbola vs line
Compare k/x and a line through two points; read inequality solution sets.
Open - Activity
Derivative and tangent
Secant slopes approach the tangent: f'(x₀) = lim Δx→0 [f(x₀+Δx)−f(x₀)]/Δx.
Guide