# Thread: Range of a parabola.

1. ## Range of a parabola.

A function is defined by $\displaystyle x^2+2x+c$ for $\displaystyle x$ is a real number. Find the value of the constant $\displaystyle c$ for which the range of f is given by $\displaystyle f(x)$≥$\displaystyle 3$

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

3. $\displaystyle x^2+ 2x+ c= x^2+ 2x+ 1- 1+ c= (x+ 1)^2+ c- 1$.

If x= -1, that value is $\displaystyle 0^2+ c- 1= c- 1$. If x is any other number, since the square is positive, the value will be greater than c- 1.

4. Sorry but I didnt understand the part "Get an expression in terms of c for the y-coordinate of the turning point"

5. Originally Posted by Punch
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.

6. From "the range of f is given by y≥3" can I say that the equation $\displaystyle x^2+2x+c$ is always positive and hence $\displaystyle b^2-4ac<0$ 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

7. Originally Posted by Punch
From "the range of f is given by y≥3" can I say that the equation $\displaystyle x^2+2x+c$ is always positive and hence $\displaystyle b^2-4ac<0$ 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 .....

8. @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

9. Originally Posted by Punch
@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?