Final answer:
The equation of the tangent line to the curve at the point (2,8) with a slope of 4 is found using the point-slope form, resulting in y = 4x which confirms that the line passes through the given point.
Step-by-step explanation:
The student is asking for the equation of the tangent line to a curve at a given point, specifically at the point (2,8) with a slope of 4. To find the equation of a tangent line, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope. In this case, the equation would be y - 8 = 4(x - 2). Simplifying this, we multiply out the right-hand side to get y - 8 = 4x - 8, and then add 8 to both sides to isolate y, giving us y = 4x. Since this must pass through the point (2, 8), we can verify it by plugging in the x value: 8 = 4(2), which simplifies to 8 = 8, confirming that the point lies on this line.
Therefore, the correct equation for the tangent line is y = 4x.