Sorry its blury. The numbers on the x axis are -1, 3, 4 and the number on the y axis is 3.
To find the solution to this problem, we must first use what we know about graphs of functions. When you try to solve for the roots of a function (the points on x where y is equal to zero, also called the x intercepts) you first try to break the formula down in pieces. For example, if you wanted to know the roots of f(x) = x^2+4x+3, you would factor it to f(x) = (x+1)(x+3), and you would know the roots/x intercepts would be -1 and -3.
In this question, the roots are already given, at -1, 3, and 4. So we just throw them into an equation really fast:
f(x)=(x+1)(x-3)(x-4)
and then we can expand it:
f(x)=(x^2-2x-3)(x-4)
f(x)=x^3 - 6x^2 + 5x + 12
Great! we have our formula. But wait, when we graph it, its gets all wonky, like its stretched out, and doesnt have the right y intercept. Well, we have to modify the formula now, with a vertical stretch. This just means we will multiply the entire formula by a certain number.
So, we know that we want the y intercept to be at 3. (Recall, the y intercept occurs when x=0) Right now, when we plug the x=0 into our equation, we end up with:
f(x)=(0)^3 - 6(0)^2 + 5(0) + 12
f(x)=12
Well, thats not right. We want it to be 3. So this is where we do our vertical stretch. We want 12 to be three, so we will have to multiply it by 0.25. !BUT! we need to make sure we do this to the entire equation.
So, our answer looks like:
f(x)=0.25[x^3 - 6x^2 + 5x + 12]
And we can expand it:
f(x)=0.25x^3 - 1.5x^2 + 1.25x + 3
And thats your answer. I dont know how well I explained, didnt use much mathimatical jargon. Hope it helps!
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