Describe the end behavior and estimate the x-and y-intercepts for each function. Estimate to the nearest tenth when necessary.
y+-2x^5+x^3
To describe the end behavior and estimate the x- and y-intercepts of the function y = -2x^5 + x^3, we can analyze the properties of the function.
1. End Behavior:
The end behavior of a function describes the behavior of the function as x approaches positive or negative infinity.
In this case, as x approaches positive infinity, both -2x^5 and x^3 increase without bound. However, the term with the higher power, -2x^5, dominates the function. Therefore, the end behavior is that y approaches negative infinity as x approaches positive infinity.
Similarly, as x approaches negative infinity, both -2x^5 and x^3 decrease without bound. Again, the term with the higher power, -2x^5, dominates the function. Therefore, the end behavior is that y approaches negative infinity as x approaches negative infinity.
In summary, the end behavior of the function y = -2x^5 + x^3 is that y approaches negative infinity as x approaches positive or negative infinity.
2. x-intercepts:
To find the x-intercepts, we set y = -2x^5 + x^3 equal to zero and solve for x:
-2x^5 + x^3 = 0
Factor out x^3:
x^3(-2x^2 + 1) = 0
From this equation, we have two possibilities:
a) x^3 = 0
In this case, x = 0 is a solution, so there is an x-intercept at (0, 0).
b) -2x^2 + 1 = 0
Solving this quadratic equation for x, we get:
-2x^2 = -1
x^2 = 1/2
x = ±√(1/2)
Therefore, there are two more x-intercepts at approximately (-0.7, 0) and (0.7, 0).
3. y-intercept:
To find the y-intercept, we set x = 0 in the equation y = -2x^5 + x^3:
y = -2(0)^5 + (0)^3
y = 0
Hence, the y-intercept is at (0, 0).
In summary, the end behavior of the function y = -2x^5 + x^3 is that y approaches negative infinity as x approaches positive or negative infinity. The x-intercepts are approximately (0, 0), (-0.7, 0), and (0.7, 0). The y-intercept is at (0, 0).