For the given function $f(x)=x^2+5x-7$:
The given function is \(f(x) = x^2 + 5x - 7\). This is a polynomial function of degree 2, commonly known as a quadratic function. In the general form \(f(x) = ax^2 + bx + c\), the degree of the polynomial is the highest power of \(x\), and the leading coefficient is the coefficient of the term with the highest power. Here, the highest power is 2, so the degree is 2. The coefficient of \(x^2\) is 1, so the leading coefficient is 1.
The end behavior of a polynomial function describes the trend of the graph as \(x\) approaches positive infinity (\(x \to \infty\)) or negative infinity (\(x \to -\infty\)). This is determined by the degree and the leading coefficient. For this function, the degree is 2 (an even number) and the leading coefficient is 1 (a positive number). Because the degree is even, both ends of the graph will behave in the same way. Because the leading coefficient is positive, the graph opens upwards, meaning that as \(x\) increases or decreases without bound, the function values increase towards positive infinity. Thus, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Understanding end behavior is fundamental in fields like economics or physics. For instance, if you model the profit of a company as a quadratic function of production quantity, the leading coefficient determines whether the profit grows boundlessly or collapses as you produce more. Recognizing that a positive leading coefficient with an even degree leads to positive infinity helps analysts predict long-term performance and potential for growth beyond the initial stages of operations.