Hi there! Your answer is f(x) = x(x+3)²(x-4)
Please see an explanation for a clear and better understanding to your problem.
Any questions about my answer or explanation can be asked through comments! :)
Step-by-step explanation:
This type of graph is called Graph of Polynomial Function. They have a degree of n where n is the positive integer. The polynomial functions can be broken into these two categorizes:
Note that they are for graph purpose only, may not appear in curriculum.
Odd-Degree functions will have graphs that start from negative infinity to positive infinity for f(x) term. Please see first and two attachments for a clear understanding.
Even-Degree functions will have graphs that start from positive infinity to positive infinity for f(x) term. Please see third and fourth attachment.
Therefore, the graph is even-degree polynomial because it starts from positive infinity where f(x) is decreasing to positive infinity where f(x) is increasing.
Next is to find a factored from of function. Factored forms are basically x-intercepts form of function. Please see the formula.
[tex]\large\boxed{f(x)=(x-a)(x-b)(x-c)}[/tex]
This is an example of factored form. The x-intercepts are at a, b and c. Just like how you solve an equation.
So what we need to do is to find x-intercepts.
The graph has x-intercepts at these following:
- x = -3 (Double)
- x = 0
- x = 4
Then we convert them into factored form including swapping from positive to negative and negative to positive. We should get:
[tex]\large{f(x)=(x+3)^2(x+0)(x-4)}\\\large{f(x)=x(x+3)^2(x-4)}[/tex]
Notice if we multiply these, we'd get 4-degree polynomial.
[ Reference to double of x = -3 ]
The graph suddenly shifts up when f(x) = 0 on x = -3. Hence, implying that x = -3 are doubled also known as (x+3)(x+3) = (x+3)²
