Solve power functions with flexible coefficients and shifts. Review derivatives, intercepts, tables, and transformed outputs. Export neat results for classwork, homework, practice, and revision.
Example function: f(x) = 2x3 + 1
| x | f(x) | f'(x) | f''(x) |
|---|---|---|---|
| -2 | -15 | 24 | -24 |
| -1 | -1 | 6 | -12 |
| 0 | 1 | 0 | 0 |
| 1 | 3 | 6 | 12 |
| 2 | 17 | 24 | 24 |
Main model: f(x) = a(x - h)n + k
First derivative: f'(x) = an(x - h)n-1
Second derivative: f''(x) = an(n - 1)(x - h)n-2
The coefficient a changes stretch and reflection. The exponent n shapes the curve. The shift h moves the graph horizontally. The shift k moves it vertically. Real-number limits apply when the base becomes negative and the exponent is not an integer.
A power function links one variable to another through an exponent. It often appears as y = axn. This calculator extends that form. It uses y = a(x - h)n + k. That makes transformed graphs easy to inspect. You can test coefficients, shifts, and exponents in one place.
Students meet power functions in algebra, precalculus, and calculus. Teachers use them for pattern analysis. Engineers use them in scaling models. Data learners use them for curve behavior checks. This page gives quick outputs for a single x value. It also builds a table over an interval. That table supports homework, revision, and classroom demonstrations.
The result block reports the function value first. It also shows the first derivative and second derivative when real outputs exist. These values help you study slope and curvature. The page also checks the domain under real-number rules. It identifies symmetry when the function is centered at the origin. For positive integer exponents, it estimates real x-intercepts too.
The coefficient a changes stretch and reflection. The exponent n controls shape and growth rate. The shift h moves the graph left or right. The shift k moves it up or down. Small changes can create very different curves. That is why interval tables are useful. You can compare outputs across many x values quickly.
Use this calculator to verify class exercises. Use it to inspect derivative patterns before graphing. Use it to build neat tables for notes. Use the export tools when you need records. The layout stays simple and clean. That keeps the focus on mathematics, not decoration.
Power functions appear in area models, volume scaling, growth comparisons, and physics relationships. A structured calculator reduces arithmetic mistakes. It also improves pattern recognition. When you change one parameter at a time, you see the role of each term clearly. This is useful during timed practice. It is also useful when preparing answer keys or checking worked examples.
Because results and tables appear together, interpretation becomes faster. You can compare input changes, graph behavior, and derivative responses without switching tools.
A power function writes one variable as a constant times a base raised to an exponent. Common forms include x2, x3, and shifted versions such as a(x - h)n + k.
Real-number power rules limit some cases. A negative base with a non-integer exponent may not give a real result. Zero raised to a negative exponent is also undefined.
The coefficient changes vertical stretch and reflection. Larger absolute values make outputs grow faster. A negative coefficient reflects the graph across the x-axis.
The exponent controls overall shape, curvature, and growth speed. Even exponents often produce matching end behavior. Odd exponents usually create opposite end behavior.
They allow graph transformations. The value h shifts the graph left or right. The value k shifts the graph up or down.
The first derivative shows slope. The second derivative shows curvature. Together, they help you study turning behavior, steepness, and local shape changes.
The table lets you inspect several x values quickly. It helps with graph sketching, checking homework, and comparing how the function changes over an interval.
Yes. After calculation, you can download the generated results as CSV or PDF. That makes classroom sharing, note keeping, and revision easier.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.