Understand that the x-intercepts of a quadratic relation are the solutions to the quadratic equation Factor a quadratic relation and find its xintercepts, and then sketch the graph Solve real-world problems by factoring a quadratic equation and finding the intercepts of the corresponding quadratic relation Determine the equation of a quadratic relation in the form y = a(x – r)(x – s) from a graph y x 2x 8 2 Set y = 0 and solve for x: x 2x 8 0 2 vertex (1,9) y ( x 4)(x 2) 0 x 4 0, x 2 0 x x 4, x 2 The x-intercepts(or zeros) of the quadratic relation 2 y ax bx c are the solutions to the quadratic equation ax bx c 0 2 If the x-intercepts r and s are found, the xcoordinate of the vertex is rs 2 The y-coordinate of the vertex is found by substituting the x-coordinate into the original equation. y 2x x 6 2 SOLUTION: y x y x 4x 4 2 SOLUTION: y x y x 3x 4 2 SOLUTION: y x 1) Two x-intercepts – two different factors leads to two solutions – graph crosses twice. 2) One x-intercept – factor is a perfect square that leads to one solution – graph just touches the x-axis. 3) No x-intercepts – cannot solve the quadratic equation by factoring – graph never touches the x-axis. y x An engineer uses the equation h d 2 25 to design an arch, where h is the height in metres and d is the horizontal distance in metres. How wide and tall is the arch? Solution: y x

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# 6.3 Graphing a Quadratic Equation by Factoring First