Graph the polynomial function. \( f(x)=4 x^{3}+7 x^{2}-10 x+2 \)

To graph the polynomial function \( f(x) = 4x^{3} + 7x^{2} - 10x + 2 \), start by identifying key features such as intercepts and turning points. The y-intercept can be easily found by substituting \( x = 0 \), which gives \( f(0) = 2 \). For x-intercepts, set the function equal to zero and solve for \( x \). Next, analyze the behavior of the function as \( x \) approaches positive and negative infinity. Because the leading coefficient (4) is positive and the degree is odd, \( f(x) \) will decrease to negative infinity as \( x \) approaches negative infinity and increase to positive infinity as \( x \) approaches positive infinity. This will help you shape the ends of the graph. Don't forget to check for local maxima and minima by finding the derivative and setting it to zero! To explore more about polynomial functions and their graphs, consider classic texts like "Calculus" by Michael Spivak or "Algebra" by Israel M. Gelfand, which provide thorough insights into behavior and manipulation of polynomials. A deep dive into these books can help deepen your understanding of functions beyond the graphing phase. For practical applications, think about how polynomials are used in various fields such as physics for projectile motion equations, in economics for determining profit maximization, or in computer graphics where polynomial curves help create smooth animations. Understanding these applications can bring more relevance and excitement to studying polynomials!