# Use Derivative to Find Quadratic Function

Use the first derivative to find the equation of a quadratic function given tangent lines to the graph of this function

Problem:a) Find the equation of a quadratic function whose graph is tangent at x = 1 to the line with slope 8, tangent at x = - 2 to the line with slope - 4 and tangent to the line y = - 8. b) Find the equation of the tangent lines at x = 1 and x = - 2. c) Graph the quadratic function obtained and the 3 tangent lines in the same coordinate system and label the tangent lines and points of tangency.
a) The slope of the tangent to the graph of a function f is related to its first derivative. Let f be the quadratic function to find to be written as
h = - b / 2a Substitute a and b by their values found above to find h = -4 / 4 = -1 The graph of the quadratic function has a vertex at (-1,8) and hence f(-1) = a(-1) ^{2} + 4(-1) + c = - 8
Solve the above equation for c to obtain c = - 6 The quadratic function f is given by f(x) = 2 x ^{2} + 4 x - 6
b) Now that we know the equations of the quadratic function, we can find the y coordinates of points of tangency of the tangent lines at x = 1 and x = -2 as follows: at x = 1, y = f(1) = 2(1) ^{2} + 4(1) - 6 = 0. The tangent line at x = 1 passes through the point (1,0).
at x = - 2, y = f(-2) = 2(-2) ^{2} + 4(-2) - 6 = - 14. The tangent line a x = -2 passes through the point (-2 , -14).
For each of the two tangent, we know the slope and a point and therefore we can find their equations. The equation of the tangent at x = 1 has slope 8 and passes through (1 , 0) and its equation is given by: y = 8x - 8 The equation of the tangent at x = -2 has slope -4 and passes through (-2 , -14) and its equation is given by: y = - 4x - 14 c) Graphs of the quadratic function and all three tangent lines.
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