Summary
Highlights
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.
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.
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.
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).