Parent functions – domain, range, symmetry, asymptotes and piecewise functions
MathematicsFunctions & GraphsAges 15–16
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Sign in to playExplore the graphs and exact key features of the parent functions y = x, x², x³, √x, ∛x, |x|, 1/x, 1/x², xⁿ for any n = p/q (negative and fractional powers included), eˣ, ln x and the floor function: domain, range, intercepts, even or odd symmetry, asymptotes, intervals of increase and decrease, and end behaviour. The Compare screen overlays families of functions (including x, x², x³ on [0, 1]), the Piecewise screen lets you define functions with two or three pieces and checks continuity at each join, and the Quiz screen asks you to identify graphs.
Lesson: Parent functions and their key features; piecewise-defined and step functions; even and odd functions
What it shows
Most graphs met later in algebra and calculus are built from a small library of parent functions. This simulation lists the exact key features of each one – domain, range, intercepts, symmetry, asymptotes, intervals of increase and decrease, and end behaviour – and marks them on the graph. Power functions xⁿ with n = p/q follow simple rules: an even q excludes negative x, a negative n excludes 0, and the parity of p decides between even and odd. For piecewise functions the range, intercepts and monotonicity are found numerically on the visible window. See also Graph transformations and Composite and inverse functions.
How to use
Choose a screen. In Gallery, pick a Function, drag P along the curve and tick Show the point at −x (symmetry test); for xⁿ set p and q. In Compare, press Powers, Roots, Reciprocals, eˣ and ln x or Compare on [0, 1], tick functions and drag the dashed line. In Piecewise, choose an Example or type f₁, f₂, f₃, then set b₁, b₂ and the piece each join belongs to. In Quiz, pick the matching function and press New graph.
Parameters you can change
- Starting screen Gallery (table of key features), Compare several functions, Piecewise functions, Identify the graph
- Starting function (Gallery) y = x (identity), y = x² (square), y = x³ (cube), y = √x (square root), y = ∛x (cube root), y = |x| (absolute value), y = 1/x (reciprocal), y = 1/x² (reciprocal square), y = xⁿ, n = p/q (power function), y = eˣ (exponential), y = ln x (natural logarithm), y = ⌊x⌋ (floor, step function)
- Numerator p of the exponent n = p/q (power function) -6–6
- Denominator q of the exponent n = p/q (power function) 1–6
- Show the point at −x to test symmetry
- Starting family (Compare) Powers: x, x², x³, Roots: x, √x, ∛x, Reciprocals: 1/x, 1/x², eˣ, ln x and the line y = x, Compare x, x², x³ on [0, 1]
- Starting piecewise example |x| in two pieces, Step function, Jump, Hole, Smooth join
Questions to explore
- Why is y = x^(2/3) defined for negative x while y = x^(3/2) is not?
- On which interval is x³ < x² < x, and why does the order reverse for x > 1?
- At a jump of a step function, which piece gives the value at the join, and how can you tell from the graph?
