How do you find a parabola with equation #y=ax^2+bx+c# that has slope 4 at x=1, slope -8 at x=-1 and passes through (2,15)?

Answer 1

The equation is #y=3x^2-2x+7#

The slope at a point is #=# the derivative.
Let #f(x)=ax^2+bx+c#
#f'(x)=2ax+b#
#f'(1)=2a+b=4#, this is equation #1#

and

#f'(-1)=-2a+b=-8#, this is equation #2#

Adding the 2 equations, we get

#2b=-4#, #=>#, #b=-2#
#2a-2=4#, from equation #1#
#a=3#

Therefore,

#f(x)=3x^2-2x+c#
The parabola passes through #(2,15)#

So,

#f(2)=3*4-2*2+c=8+c=15#
#c=15-8=7#

Finally

#f(x)=3x^2-2x+7#
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Answer 2

#y=3x^2-2x+7#

Given -

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Answer 3

To find the equation of the parabola, you need to solve for the coefficients (a), (b), and (c) in the equation (y = ax^2 + bx + c).

Given that the slope of the parabola at (x = 1) is (4), you can differentiate the equation with respect to (x) and set (x = 1). Similarly, for (x = -1), the slope is (-8). You can set up two equations using these conditions.

  1. Differentiate (y = ax^2 + bx + c) with respect to (x): [y' = 2ax + b]

  2. Substitute (x = 1) and (y' = 4): [4 = 2a(1) + b]

  3. Substitute (x = -1) and (y' = -8): [-8 = 2a(-1) + b]

  4. Solve the system of equations to find (a) and (b).

  5. Once you have (a) and (b), substitute (x = 2) and (y = 15) into the equation (y = ax^2 + bx + c) to find (c).

  6. Write the equation of the parabola as (y = ax^2 + bx + c) with the values of (a), (b), and (c) you found.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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