Final answer:
The function that would produce a slope field with parallel lines is c) y = 2x because it is a linear equation with a constant slope. The other options given are non-linear and would not create slope fields with parallel lines.
Step-by-step explanation:
The question asks which of the following functions would produce a slope field with parallel lines: a) y = x², b) y = eˣ, c) y = 2x, d) y = ln(x). A slope field, or direction field, is a graphical representation of the slopes of a differential equation at various points in the plane. It serves as a tool for visualizing the behavior of solutions to a differential equation.
Linear equations with the general form y = mx + b, where 'm' is the slope and 'b' is the y-intercept, produce graphs with constant slopes. Therefore, only functions that correspond to a linear equation will create slope fields with parallel lines since the slope is constant everywhere for such functions.
Examining the given options, the equation c) y = 2x meets this requirement. This is a linear equation without a y-intercept (b = 0), making it a special case where the line passes through the origin. In a slope field for y = 2x, every tangent line at any point on the plane will have a slope of 2, resulting in a field of parallel lines.
Let's analyze the other options briefly:
- a) y = x² is a quadratic function, which does not yield a constant slope and therefore would not create a slope field with parallel lines.
- b) y = eˣ is an exponential function, and its slope field would reflect exponentially increasing slopes and not be parallel.
- d) y = ln(x) is the natural logarithm function, and its slope field would have varying slopes dependent on the value of x, which would not be parallel.