Question

Are you ready for more? ahol amulos

100
Mai is studying the function p(x) = -x³ +25,422x² + 8x + 26. She makes a table of
values for p with x = ±1, ±5, ±10, +20 and thinks that this function has large positive
Output values in both directions on the x-axis. Do you agree with Mai? Explain your
reasoning.

Answer 1

Mai may see large positive outputs for p(x) = -x³ + 25,422x² + 8x + 26 within a limited range of x values, but the function's long-term behavior is dominated by the leading term -x³, which will result in large negative values as x grows in magnitude.

Step-by-step explanation:

A student is asking about whether the function p(x) = -x³ + 25,422x² + 8x + 26 will result in large positive output values in both directions of the x-axis. To analyze this, we consider the leading term of the polynomial, -x³. As x approaches positive or negative infinity, the behavior of p(x) is dominated by the leading term, which, in this case, is -x³. This means that for large values of x and -x, the function will have large negative values, rather than positive.

If Mai plots the values for the function at x = ±1, ±5, ±10, and 20, she may see large positive outputs for some of these x values due to the large positive quadratic term 25,422x². However, this is a localized behavior around the origin, as the cubic term will eventually dominate and make the function decrease towards negative infinity as x grows in magnitude, both positively and negatively.

Therefore, while Mai might observe large positive values for the function within the range of the x values she has chosen, the overall behavior of the function is such that it will not have large positive outputs as x approaches infinity or negative infinity.