Question

Use what you know about zeros of a function and end behavior of a graph to choose the graph that matches the function f(x)=(x-3)(x-2)(x+1)

Answer 1

The function f(x)=(x-3)(x-2)(x+1) has zeros at x=3, x=2, and x=-1, and exhibits end behavior such that as x approaches positive infinity, f(x) goes to positive infinity, and as x approaches negative infinity, f(x) goes to negative infinity.

To answer the question about the function f(x)=(x-3)(x-2)(x+1), let's analyze its zeros and end behavior.

The zeros of the function are found where the function equals zero. In this case, the zeros are x=3, x=2, and x=-1.

These are the points where the graph of the function will cross the x-axis.

To understand the end behavior of the function, consider the highest power term in the polynomial when expanded, which will be x^3, since the polynomial is of third degree.

The coefficient of x^3 will be positive (as we have an odd number of negative terms when the brackets are expanded).

Therefore, as x approaches infinity, f(x) approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity.

This indicates that the graph will rise to the right and fall to the left.

Putting it all together, the graph of f(x) will cross the x-axis at x=3, x=2, and x=-1, and the end behavior will show the right side going up towards infinity and the left side going down towards negative infinity.

Answer 2

Givens:

• 
  `f(x)=(x-3)(x-2)(x+1)`

We have a function with three roots, this means that it's a cubic function, and always increase without bound, this cubic functions increases at both ends, they don't have any restriction because they are polynomial functions. Also, we know that this function must intercept x-axis three times, because each solution represents an interception. So, these interceptions are at
`x=3;x=2;x=-1`, because these three x-values makes f(x) = 0.

So, if we graph this function we would have a behaviour like the image attached. You can observe the interception we deducted from the expression and the increasing and decreasing