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. Use the Quadratic Formula to solve each equation.The term quadratic comes from the word quadrate meaning squareor rectangular.ĩ-8The Quadratic Formula Warm Up Evaluate for x = –2, y = 3, and z = –1. Solve each quadratic equation by factoring.x= 80 x = 10 x = 20ĩ-7Completing the Square Warm Up Part I Simplify. vertex:(–2, –1) zeros:–3, –1ĩ-5Solving Quadratic Equations by Factoring Identify the vertex and zeros of the function above. y = 2x2 +2x – 8 x = 0 x = 1 (0, 2) (–2, 1)ĩ-4Solving Quadratic Equations by Graphing Warm Up 1. y = x2 + 2x + 3, when x = –1 –2 –12 2ĩ-3Graphing Quadratic Functions Warm Up Find the axis of symmetry. y = 3x + 6 Evaluate each quadratic function for the given input values. Generate ordered pairs for the function y = x2 + 2 for x-values –2, –1, 0, 1, and 2.ĩ-2Characteristics of Quadratic Functions Warm Up Find the x-intercept of each linear function. X –2 –1 0 1 2 y 6 3 2 3 6 9-1 Quadratic Equations and Functions Warm Ups Preview 9-1 Quadratic Equations and Functions 9-2 Characteristics of Quadratic Functions 9-3 Graphing Quadratic Functions 9-4 Solving Quadratic Equations by Graphing 9-5 Solving Quadratic Equations by Factoring 9-6 Solving Quadratic Equations by Using Square Roots 9-7 Completing the Square 9-8 The Quadratic Formula 9-9 The Discriminant
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