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**And the Quadratic Equation……**

Parabolas And the Quadratic Equation……

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**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.

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**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:

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**And graph the points, connecting them with a smooth curve:**

Graph of y = x2

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**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.

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Graph y = x2 + 3 The graph is shifted up 3 units from the graph of y = x2, and the vertex is (0, 3).

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Graph y = x2 - 3: The graph is shifted down 3 units from the graph of y = x2, and the vertex is (0, - 3).

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**We can also shift the vertex left and right**

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).

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**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).

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**The axis of symmetry is the line which divides the parabola into two symmetrical halves.**

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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),….

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Graph of y = x2 Graph of y = 2x2

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**Sometimes, the parabola opens downward.**

y = - (x - 2)2 + 3:

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**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

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y= -x2 X Y -3 -9 -2 -4 -1 1 2 3

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Should look like this….

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**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

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y = ½ x2 x y -3 4.5 -2 2 -1 0.5 1 3

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Should look like this….

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**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?

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y= 3x2 + 6x + 1 x y -7 106 -6 73 -5 46 -4 25 -3 10 -2 1 -1 2 3

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Should look like this….

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**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.

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