Question

Which functions have the property that the slope always increases as x increases?

1) Linear functions
2) Exponential functions
3) Quadratic functions
4) Trigonometric functions

Answer 1

Quadratic functions have the property where the slope always increases as x increases, provided the coefficient of the
`x^2` term is positive.

Step-by-step explanation:

The question is asking about the type of function where the slope always increases as the x value increases. To understand which function has this property, we need to consider the basic properties of different types of functions:

• Linear functions have a constant slope, as exemplified by the equation of a line, y = mx + b, where m is the slope. The slope does not change as x increases.
• Exponential functions, such as y =
  `a^x` (where a is a constant and a > 1), have a slope that increases as x increases, but the rate of increase changes at every point.
• Quadratic functions are represented by equations of the form y =
  `ax^2 + bx + c`, with a ≠ 0. The slope of these functions, given by the derivative 2ax + b, increases as x increases if a is positive.
• Trigonometric functions, like y = sin(x) or y = cos(x), have slopes that increase and decrease periodically, but not necessarily always increasing with increasing x.

Hence, based on these properties, quadratic functions are the ones where the slope always increases as x increases, provided that the coefficient of the
`x^2` term, a, is positive.