Final answer:
The function h(x) = 5 · 7 - 6x + 5x^2 is nonlinear because it contains the term 5x^2, indicating the highest power of x is 2. In contrast, linear functions have a degree of one, i.e., the highest power of x is x^1.
Step-by-step explanation:
Understanding Linear and Nonlinear Functions
To determine if the function h(x) = 5 · 7 - 6x + 5x2 is linear or nonlinear, we need to look at the highest power of the variable x. A linear function has a degree of one, meaning the highest power of x is x1. The given function includes the term 5x2, which indicates that the highest power of x is 2. Therefore, h(x) is a nonlinear function. A simple way to recognize a linear function is by its standard form, which is y = mx + b, where m is the slope and b is the y-intercept. Examples of linear functions include y = 55x + 75, or y = 6x + 8. All these representations show the highest power of x as x1. In contrast, terms like x2 signify a quadratic function, which is inherently nonlinear.
When working with linear functions, characteristics such as the slope and the y-intercept are often analyzed. The slope represents the rate of change, and the y-intercept is the point where the line crosses the y-axis. For instance, in a function like y = 55x + 75, the slope is 55, and the y-intercept is 75. However, with our function h(x), these concepts are not applicable due to their nonlinearity.