William W. answered • 09/19/20
Top Pre-Calc Tutor
Because this is in factored form, it is easy to see the x-intercepts, just set each binomial equal to zero.
So if x - 1 = 0 then x = 1 and if x + 3 = 0 then x = -3.
x = 1 has a multiplicity of 3 since there are three "x - 1"'s that make up the function ( it is (x - 1)3). Functions with an odd multiplicity, cross the x axis at the x-intercept.
x = -3 has a multiplicity of 2 since there are two "x + 3"'s that make up the function ( it is (x + 3)2). Functions with an even multiplicity, kiss the x axis at a single point staying on one side of the x-axis.
If we plug in very large negative numbers, we will get a very large negative answer because negative numbers to an odd power are negative and negative numbers to an even power are positive and a negative times a positive is a negative. Therefore, as x approaches negative infinity, h(x) approaches negative infinity.
If we plug in very large positive numbers, we will get a very large positive answer because positive numbers to an odd power are positive and positive numbers to an even power are also positive and a positive times a positive is a positive. Therefore, as x approaches positive infinity, h(x) approaches positive infinity.
We can also easily determine the y-intercept by plugging in x = 0. h(0) = (0 - 1)3(0 +3)2 = -9
That means the graph must look like this:
