Final answer:
To find the equation for the tangent line to the graph of the given function at (2,7), differentiate the function to find the slope, and use the point-slope form of a line to find the equation.
Step-by-step explanation:
To find the equation for the tangent line to the graph of the given function at (2,7), we need to find the derivative of the function and evaluate it at x=2. The given function is f(x) = x^2 + 3. To find the derivative f'(x), we differentiate each term: f'(x) = 2x. Evaluating f'(x) at x=2 gives the slope of the tangent line. Plugging in x=2, we get f'(2) = 2(2) = 4.
So, the slope of the tangent line at (2,7) is 4. Now, to find the equation for the tangent line, we use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope. Plugging in the values (2,7) and m=4, we get y - 7 = 4(x - 2).
Simplifying the equation gives the tangent line equation as y = 4x - 1.