Final answer:
The correct equation for the line tangent to a curve at a specific point is y - y₁ = m ( x - x₁ ), identifying the slope m and the point of tangency (x₁, y₁).
Step-by-step explanation:
The equation of the line tangent to a curve at a specific point is given by y - y₁ = m ( x - x₁ ), where (x₁, y₁) are the coordinates of the point of tangency and m is the slope of the tangent line at that point. The slope m can be calculated as the derivative of the curve's equation at the specific point.
For example, if the curve is described by y = ax + bx², its slope at any point is the derivative dy/dx = a + 2bx. Then, by knowing the coordinates of the point and the slope of the curve at that point, one can find the equation of the tangent line.
Understanding the features of a linear equation, such as y = mx + b, is essential. The coefficient of x is the slope of the line, and the constant term is the y-intercept. The slope is defined as the rise over the run between two points on the line, and the y-intercept is where the line crosses the y-axis when x = 0.