and explain the steps i just am not understanding how and after you complete the square. Thanks everyone.
your are trying to put the the equation in the vertex form :
f(x) = a(x-h)² + k
the easiest way to do that to my mind is by expanding the vertex form then identify the terms
expand the vertex form :
ax² -2axh + ah² + k
identify the terms : you have 3 equations
a= -1/2
-2ah = -1
ah² + k = 1/2
so you have the solution a = -1/2 , h = -1 and k = 1
so the vertex form is
f(x) = -1/2 (x+1)² + 1
you can try to expand it again to verify
y = (-1 / 2)x² - x + (1 / 2)
Group.
y = ((-1 / 2)x² - 1x) + (1 / 2)
Factor
y = (-1 / 2)(x² + 2x) + (1 / 2)
Add placeholders.
y = (-1 / 2)(x² + 2x + ___) + (1 / 2) + (1 / 2)(___)
Notice that the second blank is multiplied by (1 / 2) to account for what you had to add to complete the square.
Take the coefficient of the x term: 2
Divide it by 2: 2 / 2 = 1
Square it: (1)² = 1
Add 1 to both blanks.
y = (-1 / 2)(x² + 2x + 1) + (1 / 2) + (1 / 2)(1)
x² + 2x + 1 is the expanded form of a perfect square binomial.
Remember that (a + b)² = a² + 2ab + b². Apply this to what you have.
y = (-1 / 2)(x + 1)² + (1 / 2) + (1 / 2)(1)
Simplify the rest.
y = (-1 / 2)(x + 1)² + (1 / 2) + (1 / 2)
y = (-1 / 2)(x + 1)² + 1
ANSWER: y = (-1 / 2)(x + 1)² + 1 is the vertex form.
BONUS: This means that the vertex is at (-1, 1).
HINT: Remember that the vertex form is: y = a(x - h)² + k
CHECK:
y = (-1 / 2)[(x)² + 2(x)(1) + (1)²] + 1
y = (-1 / 2)(x² + 2x + 1) + 1
y = (-1 / 2)(x²) - (1 / 2)(2x) - (1 / 2)(1) + 1
y = (-1 / 2)x² - x - (1 / 2) + 1
TRUE
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Verified answer
your are trying to put the the equation in the vertex form :
f(x) = a(x-h)² + k
the easiest way to do that to my mind is by expanding the vertex form then identify the terms
expand the vertex form :
ax² -2axh + ah² + k
identify the terms : you have 3 equations
a= -1/2
-2ah = -1
ah² + k = 1/2
so you have the solution a = -1/2 , h = -1 and k = 1
so the vertex form is
f(x) = -1/2 (x+1)² + 1
you can try to expand it again to verify
y = (-1 / 2)x² - x + (1 / 2)
Group.
y = ((-1 / 2)x² - 1x) + (1 / 2)
Factor
y = (-1 / 2)(x² + 2x) + (1 / 2)
Add placeholders.
y = (-1 / 2)(x² + 2x + ___) + (1 / 2) + (1 / 2)(___)
Notice that the second blank is multiplied by (1 / 2) to account for what you had to add to complete the square.
Take the coefficient of the x term: 2
Divide it by 2: 2 / 2 = 1
Square it: (1)² = 1
Add 1 to both blanks.
y = (-1 / 2)(x² + 2x + 1) + (1 / 2) + (1 / 2)(1)
x² + 2x + 1 is the expanded form of a perfect square binomial.
Remember that (a + b)² = a² + 2ab + b². Apply this to what you have.
y = (-1 / 2)(x² + 2x + 1) + (1 / 2) + (1 / 2)(1)
y = (-1 / 2)(x + 1)² + (1 / 2) + (1 / 2)(1)
Simplify the rest.
y = (-1 / 2)(x + 1)² + (1 / 2) + (1 / 2)
y = (-1 / 2)(x + 1)² + 1
ANSWER: y = (-1 / 2)(x + 1)² + 1 is the vertex form.
BONUS: This means that the vertex is at (-1, 1).
HINT: Remember that the vertex form is: y = a(x - h)² + k
CHECK:
y = (-1 / 2)(x + 1)² + 1
y = (-1 / 2)[(x)² + 2(x)(1) + (1)²] + 1
y = (-1 / 2)(x² + 2x + 1) + 1
y = (-1 / 2)(x²) - (1 / 2)(2x) - (1 / 2)(1) + 1
y = (-1 / 2)x² - x - (1 / 2) + 1
y = (-1 / 2)x² - x + (1 / 2)
TRUE