I need help on this Calculus problem. Thanks.

Question

Write but do not solve two equations to find the x-coordinate of each point at which the line tangent to the graph of f is a) parallel to the line y=-2x+4 and b) perpendicular to the line y=-2x+4

Answer 1

If f(x)=-4x^2/9+2x/3 as it was in your previous question, f'(x)=-8x/9+2/3, and is the slope of the tangent for all x values. This slope is parallel to y=-2x+4 when -8x/9+2/3=-2, because the line and the tangent line must have the same slope of -2 to be parallel. a) The equation that has to be solved is -8x/9+2/3=-2.

b) The slope of the perpendicular is -1/(-2)=1/2, so -8x/9+2/3=1/2. This is the equation that has to be solved.

Because the equations in a) and b) are linear, there is only one solution for x, and by plugging this value into f(x) you can find the exact point on the graph where the tangent is parallel or perpendicular.