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. The term quadratic comes from the word quadrate meaning squareor rectangular.
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