Reading: Short Run Behavior: Vertex
We now explore the interesting features of the graphs of quadratics. In addition to intercepts, quadratics have an interesting feature where they change direction, called the vertex. You probably noticed that all quadratics are related to transformations of the basic quadratic function f(x) = x2.Forms of Quadratic Functions
The standard form of a quadratic function is f(x) = ax2 + bx + c The transformation form of a quadratic function is f(x) = a(x – h)2 + k The vertex of the quadratic function is located at (h, k), where h and k are the numbers in the transformation form of the function. Because the vertex appears in the transformation form, it is often called the vertex form. In the previous example, we saw that it is possible to rewrite a quadratic function given in transformation form and rewrite it in standard form by expanding the formula. It would be useful to reverse this process, since the transformation form reveals the vertex. Expanding out the general transformation form of a quadratic gives: f(x) = a(x – h)2 + k = a(x – h)(x – h) + k f(x) a(x2 – 2xh + h2) + k = ax2 – 2ahx + ah2 + k This should be equal to the standard form of the quadratic: ax2 – 2ahx + ah2 + k = ax2 + bx + c The second degree terms are already equal. For the linear terms to be equal, the coefficients must be equal: –2ah = b, so [latex]\displaystyle{h}=-\frac{{b}}{{{2}{a}}}[/latex] This provides us a method to determine the horizontal shift of the quadratic from the standard form. We could likewise set the constant terms equal to find: [latex-display]ah^{2}+k=c,\text{ so }k=c-ah^{2}=c-a\left(-\frac{b}{2a}\right)^{2}=c-a\frac{b^{2}}{4a^{2}}=c-\frac{b^{2}}{4a}[/latex-display] In practice, though, it is usually easier to remember that k is the output value of the function when the input is h, so k = f(h).Important Topics of this Section
- Quadratic functions
- Short run behavior
- Quadratic formula
Works Cited
1. From http://blog.mrmeyer.com/?p=4778, Dan Meyer, CC-BYDavid Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, " Chapter 3: Polynomial and Rational Functions," licensed under a CC BY-SA 3.0 license.
