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Warm-UpFactor.
6 minutes
1) x2 + 14x + 49
2) x2 – 22x + 121
3) x2 – 12x - 64
Solve each equation.4) d2 – 100 =
05) z2 – 2z + 1 = 06) t2 + 16 = -8t
Completing the SquareCompleting the Square Completing the SquareCompleting the SquareObjectives: •Use completing the square to solve a quadratic equation
Example 1Complete the square for each quadratic expression to form a perfect-square trinomial.
a) x2 – 10x
2b2
find
x2 – 10x + 25(x - 5)2
b) x2 + 27x
2b2
find 2
2 272
x 27x
2
x272
Practice
1) x2 – 7x 2) x2 + 16x
Complete the square for each quadratic expression to form a perfect-square trinomial. Then write the new expression as a binomial squared.
Example 2Solve x2 + 18x – 40 = 0 by completing the square.
x2 + 18x = 40
2b2
find
x2 + 18x + 81 = 40 + 81
(x + 9)2 = 121x 9 11
x = 2 or x = -20
Example 3Solve 3x2 - 6x = 5 by completing the square.
3(x2 - 2x) = 5
2b2
find
3(x2 - 2x + 1) = 5 + 3
3(x - 1)2 = 8
8x 1
3
8x 1
3
2 8(x 1)
3
PracticeSolve by completing the square.1) x2 + 10x – 24 =
0
2) 2x2 + 10x = 6
Warm-UpSolve each equation by completing the square.1) x2 + 10x + 16 = 0
2) x2 + 2x = 13
Completing the SquareCompleting the SquareCompleting the SquareCompleting the SquareObjectives: •Use the vertex form of a quadratic function to locate the axis of symmetry of its graph
Transformations
y = af(x) gives a vertical stretch or compression of fy = f(ax) gives a horizontal stretch or compression of fy = f(x) + k gives a vertical translation of f
y = f(x - k) gives a horizontal translation of f
Vertex FormIf the coordinates of the vertex of the graph of y = ax2 + bx + c, where are (h,k), then you can represent the parabola as y = a(x – h)2 + k, which is the vertex form of a quadratic function.
a 0,
Example 1Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry.
y = -6x2 + 72x - 207y = -6(x2 - 12x) - 207y = -6(x2 - 12xy = -6(x - 6)2 + 9vertex: (6,9)
axis of symmetry: x = 6
+ 36)
– 207 +216
vertex form: y = a(x – h)2 + k
Example 2Given g(x) = 2x2 + 16x + 23, write the function in vertex form, and give the coordinates of the vertex and the equation of the axis of symmetry. Then describe the transformations from f(x) = x2 to g.
g(x) = 2x2 + 16x + 23
= 2(x2 + 8x) + 23
= 2(x2 + 8x= 2(x + 4)2 - 9= 2(x – (- 4))2 + (-9)
+ 16)
+ 23
– 32
vertex: (-4,-9)
axis of symmetry: x = -4
vertex form: y = a(x – h)2 + k
PracticeGiven g(x) = 3x2 – 9x - 2, write the function in vertex form, and give the coordinates of the vertex and the equation of the axis of symmetry. Then describe the transformations from f(x) = x2 to g.
Homework