These variables are usually written as x and y, especially when you're dealing with "standardized" shapes such as a parabola. The graph of a quadratic function is a smooth, U-shaped curve that opens either upward or downward, depending on the sign of the coefficient of the x2 term. Several methods are used to find equations of parabolas given their graphs. Sketch the graph of the parabola f(x) = 3x2 – 6x – 9, labeling any intercepts and the vertex and showing the axis of symmetry. The vertex is at (3, 49): You find the x-value and then replace the x’s with 3s and simplify for the y-coordinate. A little simplification gets you the following: 5 = a(2)2 + 2, which can be further simplified to: Now that you've found the value of a, substitute it into your equation to finish the example: y = (3/4)(x - 1)2 + 2 is the equation for a parabola with vertex (1,2) and containing the point (3,5). Use the Properties of Proportions to Simplify Fractions. Example 1 Graph of parabola given x and y interceptsFind the equation of the parabola whose graph is shown below. The vertex and intercepts offer the quickest, easiest points to help with the graph of the parabola. \begin{array}{lcl} a (-1)^2 + b (-1) + c & = & 3 \\ a (0)^2 + b (0) + c & = & -2 \\ a (2)^2 + b (2) + c & = & 6 \end{array} The parabola opens upward, because 4 is positive. As a general rule, when you're working with problems in two dimensions, you're done when you have only two variables left. So you'll substitute in x = 3 and y = 5, which gives you: Now all you have to do is solve that equation for a. Another aid to use when graphing parabolas is the axis of symmetry; a parabola is symmetric about a vertical line that runs through the vertex. We can use the vertex form to find a parabola's equation. Or to put it another way, if you were to fold the parabola in half right down the middle, the vertex would be the "peak" of the parabola, right where it crossed the fold of paper. Algebra II: What Is the Binomial Theorem? The vertex is at (2.5, 4.5), and the equation of the axis of symmetry is x = 2.5. The vertex is at (1, –12), and the equation of the axis of symmetry is x = 1. Points on either side of the axis of symmetry that have the same y-value are equal distances from the axis. In real-world terms, a parabola is the arc a ball makes when you throw it, or the distinctive shape of a satellite dish. The equation of the axis of symmetry is x = h, where (h, k) is the vertex of the parabola. If you're being asked to find the equation of a parabola, you'll either be told the vertex of the parabola and at least one other point on it, or you'll be given enough information to figure those out. The graph has two x intercepts at x = − 1 and x = 2 . Since you know the vertex is at (1,2), you'll substitute in h = 1 and k = 2, which gives you the following: The last thing you have to do is find the value of a. The parabola opens upward, because 3 is positive. The easiest way to find the equation of a parabola is by using your knowledge of a special point, called the vertex, which is located on the parabola itself. Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! Sketch the graph of the parabola f (x) = – x2 + 6 x + 40, labeling any intercepts and the vertex and showing the axis of symmetry.