Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

Verified step by step guidance

1. To show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b), we first need to understand the concept of a tangent line. A tangent line to a curve at a given point is a straight line that just touches the curve at that point. For a line, the tangent line at any point on the line is the line itself.
2. The slope of the line y = mx + b is m. The derivative of y with respect to x, which gives the slope of the tangent line to the curve at any point, is also m. This is because the derivative of mx with respect to x is m, and the derivative of a constant b is 0.
3. Evaluate the derivative at the point (x₀, mx₀ + b). Since the derivative is constant and equal to m, the slope of the tangent line at any point (x₀, mx₀ + b) is m.
4. The equation of the tangent line at the point (x₀, mx₀ + b) can be written using the point-slope form: y - (mx₀ + b) = m(x - x₀).
5. Simplify the equation of the tangent line: y - mx₀ - b = mx - mx₀, which simplifies to y = mx + b. This shows that the tangent line at any point (x₀, mx₀ + b) is indeed the line y = mx + b itself.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line at that point is equal to the derivative of the function at that point. In this case, we are examining how the line itself behaves as a tangent to its own graph.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. For a linear function like y = mx + b, the derivative is constant and equal to the slope m. This means that at any point on the line, the slope of the tangent line is the same as the slope of the line itself.

Linear Functions

A linear function is a polynomial function of degree one, represented in the form y = mx + b, where m is the slope and b is the y-intercept. Linear functions graph as straight lines, and their properties, such as constant slope and direct proportionality, make them unique in that they are their own tangent lines at every point along the line.