Graph y = x3 + 6x2 + 8x and describe the end behavior. (1 point)
As the degree of the polynomial is odd and the coefficient of the leading term is positive, the end behavior of the graph will be as follows:
- As x approaches negative infinity, the y-values will also approach negative infinity.
- As x approaches positive infinity, the y-values will also approach positive infinity.
Therefore, the end behavior of the graph is that it approaches negative infinity as x approaches negative infinity, and approaches positive infinity as x approaches positive infinity.
To graph the function y = x^3 + 6x^2 + 8x, we can start by analyzing the end behavior.
End behavior refers to the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the leading term x^3 will dominate the other terms. Since x^3 is positive for positive values of x, the function will increase without bound as x goes to positive infinity.
As x approaches negative infinity, the leading term x^3 will dominate the other terms. Since x^3 is negative for negative values of x, the function will decrease without bound as x goes to negative infinity.
Therefore, the end behavior of the function y = x^3 + 6x^2 + 8x can be described as follows:
- As x approaches positive infinity, y increases without bound.
- As x approaches negative infinity, y decreases without bound.
To graph the equation y = x^3 + 6x^2 + 8x, we can follow these steps:
1. Find the x-intercepts by setting y = 0 and solving for x:
x^3 + 6x^2 + 8x = 0
Factor out an x:
x(x^2 + 6x + 8) = 0
The x-intercepts are x = 0 and the solutions to the quadratic equation x^2 + 6x + 8 = 0.
2. Find the y-intercept by setting x = 0:
y = 0^3 + 6(0)^2 + 8(0) = 0
The y-intercept is (0, 0).
3. Determine the end behavior by looking at the leading coefficient of the highest power of x. In this case, the leading coefficient is 1, which is positive.
When the leading coefficient is positive and the power of x is odd (in this case, 3), the graph will have a similar shape as a cubic function. The ends of the graph will point in opposite directions, with the left end pointing downwards (negative infinity) and the right end pointing upwards (positive infinity).
So, the end behavior of the graph y = x^3 + 6x^2 + 8x is as follows:
As x approaches negative infinity, y approaches negative infinity.
As x approaches positive infinity, y approaches positive infinity.
Thus, the graph of y = x^3 + 6x^2 + 8x will have a "U" shape, with the left side of the U pointing downwards and the right side pointing upwards.