SOLUTION: the graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine the expression for the function.

Algebra ->  Equations -> SOLUTION: the graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine the expression for the function.      Log On


   

Question 206260: the graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine the expression for the function.
Found 2 solutions by Marth, Theo:
Answer by Marth(57) About Me  (Show Source):
You can put this solution on YOUR website!
The base function is y=x%5E2
"a range consisting of all numbers less than or equal to 4"
The parabola must open downwards because of less than or equal.
y=-x%5E2
But, y=-x%5E2 has a range of (-infinity, 0]. So you need to shift it up 4.
y=-x%5E2%2B4


"x-intercepts -1 and 3"
The normal x intercepts for y=-x%5E2%2B4 would be -2, 2.
You need to shift the parabola to the right 1 to get -1, 3.
y=-%28x-1%29%5E2%2B4

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your x intercepts are -1 and 3 and this is a quadratic equation.
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this means that (x+1)*(x-3) = 0 because that is the quadratic equation that would yield these 2 x intercepts.
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multiplying them out we get x%5E2+-+2x+-+3+=+0
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that's our equation.
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a limitation is that the range of our function can't be greater than 4.
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Since y represents the range of our function, this means y has to be less than or equal to 4.
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since our equation is y = f(x) = x%5E2+-+2x+-+3, this means that x%5E2+-2x+-+3+=+4 would represent the maximum value that y could be.
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in order to restrict the range, we have to restrict the domain because the range is dependent on the domain.
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we need to find the x values for when y = 4
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if we subtract 4 from both sides of the equation of x%5E2+-2x+-+3+=+4 we get x%5E2+-2x+-+7+=+0
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if we find the roots of this equation, we should be able to find the x values for when y = 4.
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solving the equation of x%5E2+-2x+-+7+=+0 using the quadratic formula of %28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 yields the following answers:
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x = 3.828427125 or x = -1.828427125
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we got this in the following manner:
a = 1
b = -2
c = -7
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2a = 2
-b = 2
sqrt%28b%5E2+-+4ac%29 = sqrt%284-%284%2A1%2A%28-7%29%29%29 = sqrt%284%2B28%29 = sqrt%2832%29
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x = %28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 becomes x = %282+%2B-+sqrt%2832%29%29%2F2 which becomes x = 3.828427125 or x = -1.828427125.
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The graph of our equation is x%5E2+-+2x+-+3 with the restriction that -1.828427125+%3C=+x+%3C=+3.828427125.
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a graph of our equation would look like this:
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you can see that the lower limit for x will be around -1.828... and the upper limit for x will be around +3.828... and that anything in between is good.
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those horizontal and vertical lines are just there to let you see the x values and the y values easier. they are not part of the quadratic equation.