![]() To get from the third to the fourth, distribute and collect like terms. To get from the second to the third, square the binomial in parentheses. To get from the first to the second, complete the square. Does this path surprise you? Please continue reading the text below the graph to explore why this behavior occurs.Īll of the equations shown in the bottom of the applet are equivalent equations. Pay attention of the path that the point V travels as you move the b slider back and forth. Use the blue slider to vary the value of the linear term b. Changing c translates the graph vertically by adding a constant value to all y-coordinates on the graph, as shown by the Vertex Form of the equation. Notice how the graph slides straight up or down, without changing its shape at all. Use the green slider to vary the value of c, the constant term. a is the vertical dilation factor for this function, as shown by the Vertex Form of the equation. Notice how the graph becomes wider or taller, and reflects vertically about the x-axis when a becomes negative. Use the red slider to vary the value of a, the coefficient of the squared term. ![]() The first equation is in "Standard Form", the second in "Vertex Form" (start with the Standard Form, then complete the square), and the remaining ones expand the Vertex Form for reasons that will be explained below. Note that the vertex of the parabola is identified on the graph as point V, with its coordinates shown.īelow the three sliders are a series of equivalent equations, each of which describes the graph being shown. You may click and drag them left or right to alter the value of each coefficient, and the graph will change to reflect the new value. The graph below contains three sliders, one for each coefficient. However, changing the value of b causes the graph to change in a way that puzzles many. Changing either a or c causes the graph to change in ways that most people can understand after a little thought. ![]() A quadratic equation in "Standard Form" has three coefficients: a, b, and c. ![]()
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