To find the equation of the line of symmetry,it will always be y = c, where c is always the x-value of thevertex (x, y). Remember from earlier lessons that vertical lines are alwaysin the form x = c. Since the line of symmetry will alwaysbe a vertical line in all of our parabolas, the general formulafor the line will be x = c. This vertical line is called the line of symmetry or axis of symmetry. If youdraw a vertical line through the vertex, it will split the parabolain half so that either side of the vertical line is symmetricwith respect to the other side. The lowest pointon the graph is (1, -1) and is called the vertex.
Here, we see again that the x- and y-intercepts are both (0,0), as the parabola crosses through the origin. What are the x-and y-intercepts? What is the lowest point onthe graph? Let's substitute the same values in for x as we did in thechart above and see what we get for y. So, we would have the equation, y = x 2-2x. Let's trying graphing another parabola where a = 1, b = -2and c = 0. Parabolas in the standard from y = ax 2 + bx +c. The secondform is called the vertex-form or the a-h-k form,y = a(x - h) 2 + k. The first form is calledthe standard form, y = ax 2 + bx + c. The general shape of a parabola is the shape of a "pointy"letter "u," or a slightly rounded letter, "v."You may encounter a parabola that is "laying on it's side,"but we won't discuss such a parabola here because it is not afunction as it would not pass the Vertical Line Test. What is the lowest point on the graph? Can you tell if thereare any high points on the graph? Where does it cross the x- andy-axes? Going from left to right like you would read, where doesthe graph seem to be decreasing and where does it increase? Click here for the answers. Plot the graph on your own graph paper and make sure that youget the same graph as depicted below. So, let's try substituting values in for x and solvingfor y as depicted in the chart below. Remember, if you are not surehow to start graphing an equation, you can always substitute anyvalue you want for x, solve for y, and plot the correspondingcoordinates. We said thatthe graph of y = x 2 was a function because it passedthe vertical line test. We talked a little bit about this graphwhen we were talking about the Vertical Line Test. What about a quadratic equation? What are the characteristicsof a quadratic function? Well, if we look at the simplest casewhen a = 1, and b = c = 0, we get the equation y = 1x 2or y = x 2. Note thatif a = 0, the x 2 term would disappear and we wouldhave a linear equation! Thus, the standardized form of a quadratic equation is ax 2+ bx + c = 0, where "a" does not equal 0. Simply, the three terms include one that hasan x 2, one has an x, and one term is "by itself"with no x 2 or x. Normally, we see thestandard quadratic equation written as the sum of three termsset equal to zero. So, for our purposes, we willbe working with quadratic equations which mean that the highestdegree we'll be encountering is a square. In an algebraic sense, the definition ofsomething quadratic involves the square and no higher power ofan unknown quantity second degree. Similarly, one of the definitions of the termquadratic is a square. ( y ± d ) = a ( x ± f ) 2 (y \pm d) = a(x \pm f)^ (x + 2)(x - 5) y = 5 1 ( x + 2 ) ( x − 5 )Īnd that's all there is to it! Those are the two most important methods for finding a quadratic function from a given parabola.The term quadratic comes from the word quadrate meaning squareor rectangular. The vertex formula is as follows, where (d,f) is the vertex point and (x,y) is the other point: With the vertex and one other point, we can sub these coordinates into what is called the "vertex form" and then solve for our equation. In order to find a quadratic equation from a graph using only 2 points, one of those points must be the vertex. In order to find a quadratic equation from a graph, there are two simple methods one can employ: using 2 points, or using 3 points.
QUADRATIC FUNCTION GRAPH HOW TO
Now let's get into solving problems with this knowledge, namely, how to find the equation of a parabola! How to Find a Quadratic Equation from a Graph: The more comfortable you are with quadratic graphs and expressions, the easier this topic will be! But, before we get into these types of problems, take a moment to play around with quadratic expressions on this wonderful online graphing calculator here. In this article, the focus will be placed upon how we can develop a quadratic equation from a quadratic graph using a couple different methods.
There are so many different types of problems you can be asked with regards to quadratic equations. Sample graph of a simple quadratic expression