Quadratic Functions Transformations
Explore how algebraic transformations dynamically affect the graph of a quadratic function. Enter a base quadratic function \(f(x)\), select a transformation type, and observe both graphs plotted simultaneously. Analyze how reflections, stretches, compressions, and absolute values shift the vertex, intercepts, and orientation of the parabola.
Understanding Quadratic Transformations
This interactive tool demonstrates how each transformation acts on a quadratic function, both algebraically and visually:
- \(-f(x)\): Reflects the graph vertically across the x-axis, flipping the parabola upside down.
- \(f(-x)\): Reflects the graph horizontally across the y-axis.
- \(-f(-x)\): Reflects the graph through the origin (a 180° rotation), combining both reflections.
- \(|f(x)|\): Takes the absolute value of the output. Any part of the parabola below the x-axis is reflected above it.
- \(f(|x|)\): Takes the absolute value of the input. The branch for \(x \ge 0\) is kept, and the graph is mirrored across the y-axis to produce the \(x < 0\) branch.
- \(f(x/2)\): Horizontally stretches the graph by a factor of 2 (the parabola becomes wider).
- \(f(2x)\): Horizontally compresses the graph by a factor of \(1/2\) (the parabola becomes narrower).
- \(f(x)+f(-x)\): The even part of \(f\), doubled. Because \(f(x)+f(-x)\) is unchanged when \(x\) is replaced by \(-x\), the result is always symmetric about the y-axis.
- \(f(x)-f(-x)\): The odd part of \(f\), doubled. Replacing \(x\) by \(-x\) negates the expression, so the result is symmetric about the origin.
Frequently Asked Questions
What is the difference between \(-f(x)\) and \(f(-x)\)?
\(-f(x)\) negates the output, reflecting the graph across the horizontal x-axis. \(f(-x)\) negates the input, reflecting the graph across the vertical y-axis. For a parabola, these produce visibly different curves unless the parabola is symmetric about both axes (which only the constant zero function is).
Why does \(f(|x|)\) always produce a graph symmetric about the y-axis?
Because \(|x| = |-x|\), the function returns the same value for \(x\) and \(-x\). The right branch (\(x \ge 0\)) is kept and mirrored to the left, so the result is always even.
What is meant by "even combination" \(f(x)+f(-x)\)?
For any function \(f\), the expression \(E(x) = f(x)+f(-x)\) is unchanged when \(x\) is replaced by \(-x\), so \(E\) is an even function. It is twice the even part of \(f\). Likewise, \(O(x) = f(x)-f(-x)\) satisfies \(O(-x) = -O(x)\), so it is odd (twice the odd part of \(f\)).