And the Quadratic Equation……. Parabola - The shape of the graph of y = a(x - h) 2 + k Vertex -...
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PARABOLASAnd 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. 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.
Axis of Symmetry
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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
0 0
1 -1
2 -4
3 -9
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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
0 0
1 0.5
2 2
3 4.5
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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 + 1x y
-7 106
-6 73
-5 46
-4 25
-3 10
-2 1
-1 -2
0 1
1 10
2 25
3 46
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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.