Power functions
About this app
The graph of f(x)=a·xⁿ with sliders for n and a. Even n → axis-symmetric, odd → point-symmetric; n<0 → hyperbola. Function family, symmetry highlight, notable points.
Basic shapes
A power function has the form f(x) = a·xⁿ. The exponent n sets the basic shape: for even n such as 2 or 4 the graph is symmetric about the y-axis, for odd n such as 1 or 3 it is symmetric about the origin. Negative n gives a hyperbola with a pole at x = 0. The factor a stretches or compresses the graph and reflects it in the x-axis when a < 0.
What the app shows
Sliders for the exponent n, the factor a and the shifts c and d change the graph of f(x) = a·(x − c)ⁿ + d live. The app can show a whole family of functions, highlight the symmetry and draw a grid. A values panel lists the function term, the type of symmetry, the domain and key points.
Try it yourself
Increase n step by step from 1 to 6 and note that with a = 1 every graph passes through (1 | 1), and for even n also through (−1 | 1). The larger n, the flatter the graph between −1 and 1 and the steeper outside. With c and d you can see the centre of symmetry move together with the graph.
Frequently asked questions
- When is a power function symmetric about the y-axis?
- f(x) = a·xⁿ is symmetric about the y-axis for even exponents, because f(−x) = f(x). For odd exponents it is symmetric about the origin, since f(−x) = −f(x).
- What does the graph of a power function with a negative exponent look like?
- For n < 0, f(x) = a·xⁿ = a / x^|n| is a hyperbola. It is undefined at x = 0 and has a pole there; for large |x| the graph approaches the x-axis.
- Which points do all power functions share?
- With a = 1, every graph of f(x) = xⁿ passes through (1 | 1). For even n it also passes through (−1 | 1), for odd n through (−1 | −1). For positive n the origin lies on the graph as well.
- What can I set in the app?
- You change the exponent, the stretch factor and the shifts with sliders and can show a family of functions, the symmetry and a grid. Symmetry, domain and key points are updated continuously.
Subject: School mathematics (Gymnasium)
More from School mathematics (Gymnasium)
- Tree diagram & path rules
- Binomial identities through geometry
- The number π
- Unit circle & sine curve
- Exponential growth & decay
- Ulm’s glass pyramid
- Equation balance
- Half-life & decay
A VisuApp by heyprof: interactive, free in your browser, no sign-up. What is a VisuApp? · All apps