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Quadratic equations function in the same manner as linear equations in that you have a particular input resulting in a particular output. You plot points on the graph in the same manner as you do linear equations. The main difference is that the graph of a quadratic forms a curve, called a parabola, instead of a straight line. You will not need to know the quadratic formula for the ISEE. Quadratic equations take the following form: Graphs of Quadratic Equations How the Coefficient Affects the Shape of the Graph The coefficient a determines the direction and width of the parabola: A larger number creates a skinnier parabola A number between -1 and 1 creates a fatter parabola. If a is positive the parabola curves up (positive = smiley face) If a is negative the parabola curves down (negative = sad face). We know that c is the y-intercept because inputting 0 for x2 and x gives: y = a(0)2 + b(0) + c = 0 + 0 + c = c. In graph (a ) below, the coefficient a is positive and the parabola curves up (in a “smile”). In graph (b) the coefficient a is negative and the parabola curves down (in a “frown”). Finding the X-Intercepts Aside from knowing what a quadratic equation will look like on a graph, you will need to know how to find its x-intercepts and its vertex (the highest or lowest point of a parabola). To find a quadratic equation’s x-intercepts, the quadratic equation must be set equal to zero (0): The reason for this is that for any coordinate that falls on the x-axis, the value of y will be zero (ex. (3, 0) or (‐3, 0)). This is also true for solving for x for quadratic equations, since it is the same thing. If your quadratic equation is set to anything other than zero, you must manipulate the quadratic equation to equal zero. To find the intercepts, set the equation to equal zero, factor out the quadratic equation, then use the zero product property to solve for x. Example of finding the X-Intercepts Finding the Vertex The vertex of a parabola is the “tip” of the parabola, where it changes directions. The xy-coordinate of a vertex is (h, k). To find the vertex of a quadratic equation, use the following equation to find h: Once you have solved for h, you can input this value as the x input in the quadratic equation. The value of y once you solve the equation is the value k. Sometimes a quadratic equation will be presented in the following format: y = a(x - h)2 + k In this case, it is easy to see that (h, k) is the vertex. Example of finding the Vertex Keep in mind, when a quadratic is positive the vertex is the lowest point on the parabola, and when a quadratic is negative the vertex is the highest point on the parabola. Practice Questions Answers to Practice Questions x-intercepts = 2, 6. Vertex = (4, -4) x-intercepts = 2, -4. Vertex = (-1, 9) x-intercepts = -1, -3. Vertex = (-2, -2) x-intercepts = -2, -5. Vertex = (-3.5, 2.25) C B A B
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