# how to find the equation of a parabola from a graph

5 = a (2)2 + 2, which can be further simplified to: 5 = a (4) + 2, which in turn becomes: 3 = a (4), and finally: a = 3/4. Following are answers to the practice questions: The only intercept is at (0, 0). In either formula, the coordinates (h,k) represent the vertex of the parabola, which is the point where the parabola's axis of symmetry crosses the line of the parabola itself. Examples are presented along with their detailed solutions and exercises. With all those letters and numbers floating around, it can be hard to know when you're "done" finding a formula! The parabola opens downward because –2 is negative. Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! You're told that the parabola's vertex is at the point (1,2), that it opens vertically and that another point on the parabola is (3,5). Graph of y = x² - The Simplest Parabola The simplest parabola, y = x² The vertex and intercepts offer the quickest, easiest points to help with the graph of the parabola. eval(ez_write_tag([[250,250],'analyzemath_com-medrectangle-3','ezslot_1',320,'0','0']));Solution to Example 1The graph has two x intercepts at $$x = - 1$$ and $$x = 2$$. Sketch the graph of the parabola f(x) = –2x2 + 10x – 8, labeling any intercepts and the vertex and showing the axis of symmetry. -- math subjects like algebra and calculus. 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. The intercepts are at (0, 3), (3, 0), and (–3, 0). These variables are usually written as x and y, especially when you're dealing with "standardized" shapes such as a parabola. The equation of the axis of symmetry is x = h, where (h, k) is the vertex of the parabola. To do that choose any point (x,y) on the parabola, as long as that point is not the vertex, and substitute it into the equation. The quadratic equation is sometimes also known as the "standard form" formula of a parabola. The idea is to use the coordinates of its vertex ( maximum point , or minimum point ) to write its equation in the form $$y=a\begin{pmatrix}x-h\end{pmatrix}^2+k$$ (assuming we can read the coordinates $$\begin{pmatrix}h,k\end{pmatrix}$$ from the graph) and then to find the value of the coefficient $$a$$. \)Simplify and rewrite as$$Let's do an example problem to see how it works. 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. Sketch the graph of the parabola f(x) = 4x2, labeling any intercepts and the vertex and showing the axis of symmetry. Solution to Example 3The equation of a parabola with vertical axis may be written as\( y = a x^2 + b x + c$$Three points on the given graph of the parabola have coordinates $$(-1,3), (0,-2)$$ and $$(2,6)$$. Next, substitute the parabola's vertex coordinates (h, k) into the formula you chose in Step 1. Points on either side of the axis of symmetry that have the same y-value are equal distances from the axis.
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.

Your very first priority has to be deciding which form of the vertex equation you'll use. 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. Sketch the graph of the parabola f (x) = – x2 + 6 x + 40, labeling any intercepts and the vertex and showing the axis of symmetry. Example 1 Graph of parabola given x and y interceptsFind the equation of the parabola whose graph is shown below. Solution to Example 4The parabolic reflector has a vertex at the origin $$(0,0)$$, hence its equation is given by$$y = \dfrac{1}{4p} x^2$$The diameter and depth given may be interpreted as a point of coordinates $$(D/2 , d) = (1.15 , 0.35)$$ on the graph of the parabolic reflector. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. Sketch the graph of the parabola f(x) = –x2 + 6x + 40, labeling any intercepts and the vertex and showing the axis of symmetry. As you can see, the y-intercept is (0, 40); you can find it by letting all the x’s equal 0 and simplifying.