Summary
Highlights
Introduction to Quadratic Function Forms00:00:01
The video introduces two forms for writing quadratic functions: the standard form (y = ax² + bx + c) and the vertex form (y = a(x - h)² + k). The remainder of the video demonstrates how to derive a quadratic function using one of these forms based on given conditions: a table of values, a graph, or known zeros.
Finding Quadratic Function from a Table of Values00:00:27
To find a quadratic function from a table of values, first check if the x-values are consecutive. Calculate the first and second differences of the y-values. If the second differences are constant, it confirms a quadratic function. To find 'a', divide the second difference by 2. 'c' is the y-intercept (where x=0). Substitute a known (x,y) point and the found 'a' and 'c' values into y = ax² + bx + c to solve for 'b'. Finally, write the quadratic function using the calculated a, b, and c values.
Finding Quadratic Function from a Graph (Vertex Form)00:03:44
When given a graph, use the vertex form y = a(x - h)² + k. The vertex (h, k) is directly identifiable from the graph. Choose any other point (x, y) from the graph and substitute the values of h, k, x, and y into the vertex form equation. Solve the equation for 'a'. Once 'a', 'h', and 'k' are known, write the quadratic function in vertex form.
Finding Quadratic Function from Zeros (Factor Form)00:05:48
Given the zeros of a quadratic function (e.g., x = -2 and x = 5), these can be expressed as factors (x + 2) and (x - 5). Multiply these factors together to get the quadratic equation, and then replace '0' with 'y' to form the quadratic function. An alternative method uses the sum and product of roots: the sum of roots is -b/a and the product is c/a. For a quadratic where a=1, the equation is y = x² - (sum of roots)x + (product of roots).