Glossary Checklist FAQ

Transforming the Basic Quadratic Function



The most basic quadratic function is \(f(x) = x^2 \). The basic quadratic can be transformed into \(f(x) = a(x - p)^2 + q \) by multiplying by \(a\) as well by adding values of \(p\) and \(q\), which translate (move) each point \(p\) units right or left and \(q\) units up or down.

As mentioned before, when \(a < -1\) or \(a > 1\) , the graph is made narrower, and when \(-1 < a < 1\), the graph is made wider. A negative \(a\) value will cause the graph to open downward and a positive \(a\) value will cause the graph to open upward.

If \(p\) is positive, the parabola is translated to the right; if \(p\) is negative, the parabola is translated to the left. If \(q\) is positive, the parabola is translated up; if \(q\) is negative, the parabola is translated down.

When transforming the graph, it is important to apply the stretch caused by \(a\) first, followed by the translations caused by \(p\) and \(q\).