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Math Help - Range of a parabola.

  1. #1
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    Range of a parabola.

    A function is defined by x^2+2x+c for x is a real number. Find the value of the constant c for which the range of f is given by f(x) 3
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  2. #2
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    Quote Originally Posted by Punch View Post
    A function is defined by x^2+2x+c for x is a real number. Find the value of the constant c for which the range of f is given by f(x) 3
    Get an expression in terms of c for the y-coordinate of the turning point, equate it to 3 and solve for c.
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  3. #3
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    x^2+ 2x+ c= x^2+ 2x+ 1- 1+ c= (x+ 1)^2+ c- 1.

    If x= -1, that value is 0^2+ c- 1= c- 1. If x is any other number, since the square is positive, the value will be greater than c- 1.
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    Sorry but I didnt understand the part "Get an expression in terms of c for the y-coordinate of the turning point"
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    Quote Originally Posted by Punch View Post
    Sorry but I didnt understand the part "Get an expression in terms of c for the y-coordinate of the turning point"
    From post #3, the "expression in terms of c for the y-coordinate of the turning point" is c - 1.
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    From "the range of f is given by y≥3" can I say that the equation x^2+2x+c is always positive and hence b^2-4ac<0 and solve for the range of c?

    I dont know if this is the right way to solve, but i really dont understand how to solve the question even after all of your help...

    i am afraid i might need more explanation, i am sorry for wasting your time
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    Quote Originally Posted by Punch View Post
    From "the range of f is given by y≥3" can I say that the equation x^2+2x+c is always positive and hence b^2-4ac<0 and solve for the range of c?

    I dont know if this is the right way to solve, but i really dont understand how to solve the question even after all of your help...

    i am afraid i might need more explanation, i am sorry for wasting your time
    You should know that the range of a positive parabola is all the values of y equal to or greater than the y-coordinate of the turning point.

    In your question you have a positive parabola and it has been pointed out to you that the y-coordinate of the turning point is c - 1. Therefore .....
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    @mr. fantastic, after reading your post, i came up with this solution by using calculus to solve..

    dy/dx=2x+2

    turning point, 2x+2=0
    x=-1

    sub x=-1, y=1-2+c

    turning point, y=3

    3=1-2+c

    c=4
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  9. #9
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    Quote Originally Posted by Punch View Post
    @mr. fantastic, after reading your post, i came up with this solution by using calculus to solve..

    dy/dx=2x+2

    turning point, 2x+2=0
    x=-1

    sub x=-1, y=1-2+c

    turning point, y=3

    3=1-2+c

    c=4
    That's good.

    But since this question has been posted in the PRE-calculus subforum, the expectation of members is that it will be solved without using calculus. That's why post #3 completes the square. Do you know how to complete the square and do you know how to read the coordinates of the turning point from y = a(x - h)^2 + k?
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