Understanding Quadratic Dilation
A quadratic dilation changes the width, height, or position of a parabola. The original equation may look simple. Yet every scale factor changes important features. This calculator uses coefficients, scale factors, and a center point. It then builds the transformed equation in standard form. The tool also reports the vertex, discriminant, roots, and axis of symmetry.
Why Dilation Matters
Dilation helps students compare similar parabolas. A horizontal factor changes how quickly x values move from the center. A vertical factor changes y values around the center. When the factor is greater than one, the graph stretches. When it is between zero and one, the graph compresses. Negative factors can reflect the curve. This makes the calculator useful for algebra, graphing, and transformation checks.
Input Choices
The calculator accepts a, b, and c from the standard form. It also accepts horizontal and vertical scale factors. The center point controls where the dilation is measured. A center at zero gives the common origin based transform. A different center creates a more advanced result. The precision box controls rounded output. This helps match homework, notes, or classroom examples.
Result Details
The result area appears above the form after submission. It shows the new equation first. Then it lists the transformed vertex and axis. It also compares discriminants and roots when real roots exist. The step notes explain each stage. The example table gives quick test data. CSV and PDF buttons save the same calculation for later review.
Study Benefits
Quadratic dilation is easier when each changed coefficient is visible. This page separates the original curve from the transformed curve. It also shows the formulas used. Students can test many values quickly. Teachers can prepare examples with consistent output. The final equation can be copied into a graphing tool. Repeating this process builds strong transformation skills and better equation sense.
Common Mistakes
Many errors come from mixing horizontal and vertical factors. Horizontal dilation divides the input distance before the original function is evaluated. Vertical dilation multiplies the output distance after evaluation. The order matters. Always enter nonzero horizontal factors. Check the center point carefully. A small center change can alter every coefficient in the final standard equation during the calculation process.