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## Parabolas

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**Parabolas**And the Quadratic Equation……**Terms:**• Parabola - The shape of the graph of y= a(x - h)2 + k • Vertex - The minimum point in a parabola that opens upward or the maximum point in a parabola that opens downward. • Quadratic Equation - An equation of the form ax2 + bx + c = 0, where a ≠ 0, and a, b, and c are real numbers. • Axis of Symmetry - The line which divides the parabola into two symmetrical halves.**Graphing y= x2**• Is this a linear function? • What does the graph of y = x2 look like? • To find the answer, make a data table:**And graph the points, connecting them with a smooth curve:**Graph of y = x2**The shape of this graph is a parabola.**The parabola does not have a constant slope. In fact, as x increases by 1, starting with x = 0, y increases by 1, 3, 5, 7,…. As x decreases by 1, starting with x = 0, y again increases by 1, 3, 5, 7,…. In the graph of y = x2, the point (0, 0) is called the vertex.**Graph y = x2 + 3**The graph is shifted up 3 units from the graph of y = x2, and the vertex is (0, 3).**Graph y = x2 - 3:**The graph is shifted down 3 units from the graph of y = x2, and the vertex is (0, - 3).**We can also shift the vertex left and right. Look at the**graph of y = (x + 3)2 The graph is shifted left 3 units from the graph of y = x2, and the vertex is (- 3, 0).**Observe the graph of y = (x - 3)2:**The graph is shifted to the right 3 units from the graph of y = x2, and the vertex is (3, 0).**The axis of symmetry is the line which divides the parabola**into two symmetrical halves. Axis of Symmetry**As well as shifting the parabola up, down, left, and right,**we can stretch or shrink the parabola vertically by a constant. • Data table for the graph of y = 2x2: • Here, the y increases from the vertex by 2, 6, 10, 14,…; that is, by 2(1), 2(3), 2(5), 2(7),….**Graph of y = x2**Graph of y = 2x2**Sometimes, the parabola opens downward.**• y = - (x - 2)2 + 3:**What can we find from y= -x2 ?**• Which way will the parabola open? • The negative a value indicates - Down • Where is the vertex? • Make a table of values to be sure**What can we find from y= ½ x2 ?**• Which way will the parabola open? • The a value is positive - Up • Where is the vertex? • What is the step pattern? • Make a table of values to be sure**What can we find from y = 3x2 + 6x + 1?**• Which way will the parabola open? • The a value is positive – Up • What will the vertical stretch be? • What will the step pattern be? • What is the y-intercept?**Forms of the Quadratic equation:**• Standard form • y = ax2 + bx + c where c is the y-intercept • Vertex form • y = a (x - h)2 + k where (h,k) is the vertex • Factored form • y = a (x - s)(x – t) where s and t are the zeros • For the same parabola, the quadratic equation in any form will have the SAME a value – which indicates the direction of opening and the vertical stretch.