Question

HELPP.
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
(c) Find the y-intercept.

Answer 1

\[f(x) = x ^{2} (x+2)\]

Answer 2

Alright, well first we need to determine the degree of the function, meaning the highest power involved. If you were to multiply that out, you would see that the highest power would be a 3. When you have a function of an odd degree, the function has opposite end behavior on the graph. In an odd function, if the coefficient of the highest power is positive, the graph comes from the bottom left and leaves out the top right. So that is the case in your equation. As for the x-intercepts, you need to set each factor of your equation equal to 0. So one factor is x^2 and the other factor is (x+2). Setting x^2 = to 0 gives you x= 0 and (x+2) = to 0 gives you x = -2. So 0 and-2 are your x-intercepts. Y-intercepts are found by replacing x with 0 and solving for y. If you set x = 0, the whole function = 0, meaning our y-intercept is just 0. So now in regards to touching or crossing, we want to refer to multiplicity. Essentially, multiplicity is the amount of times you have the same x-intercept. Above, we had x^2 and (x+2) as factors. So one of our x intercepts was 0, which occured twice because of the power of 2. Meaning the intercept at 0 has a multiplicity of 2. The x = -2 intercept only occured once, so its multiplicty is 1. When you have an even multiplicty, the graph will touch at that intercept and turn around. When you have an odd multiplicity, the graph will cross at that intercept. So for quick recap
*function of degree 3 and positive lead coefficient means graph comes from bottom left and exits through thetop right
*setting each factor equal to 0 gives x-intercepts of 0 and -2
*Replacing x with 0 leaves a y-intercept of 0, too
*The x-intercept of 0 has a multiplicty of 2, meaning the graph touches and turns around at this point. The x-intercept at -2 has a multiplicty of 1, meaning itll cross at that point.