# What is the function graphed, f(x)=? Hint: you may write the function as f(x)=a(x-b)(x-c)(x-d), where b,c, and d are integers and a is a fraction

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) that has been graphed can be determined by looking at the intercepts. From the graph the curve intersects the x-axis at the points (3, 0) and (-5, 0) and it intersects the y-axis at (0,-1).

f(x) can be written as f(x) = a(x-b)(x-c)(x-d).

The slope of the tangent drawn to the graph at a point where x = a is given by f'(a).

From the graph, the slope of the tangent to the curve is 0 at two points (0, -1) and (-5, 0)

Now, substitute the coordinates of the points that it passes through.

As the graph goes through (-5, 0) and (3, 0)

f(x) = a*(x - 3)(x + 5)*(x - d)

= a*x^3+a*(2-d)*x^2+a*(-2*d-15)*x+15*d*a

f'(x) = 3*a*x^2+(4*a-2*a*d)*x-2*a*d-15*a

At (-5, 0), f'(-5) = 0

0 = 3*a*25+(4*a-2*a*d)*(-5)-2*a*d-15*a

=> 0 = 8*a*d+40*a

=> 0 = 8a*(d + 5)

As a `!=` 0, d = -5

As the graph passes through (0, -1)

-1 = a*(0 - 3)(0 - 5)(0 - 5)

=> a = 1/75

The function f(x) = (1/75)*(x - 3)(x + 5)^2 = (x^3+7*x^2-5*x-75)/75

The graph of this function is:

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