Question

Which of these equations has a graph that is tangent to the x-axis at one of its intercepts? Choose all the polynomials that have this property.

a. y = x^2 + 6x + 9
b. y = –x^3
c. y = (x + 2)(x + 6)^2
d. y = (x – 5)(x + 3)(x + 1)(x – 5)
e. y = x^4

Answer 1

Another way to put this: the x-axis is the tangent line to the graph of one or more of these functions. The point of tangency is also the x-intercept (or one of those intercepts).

Example: y=x^4 has a graph which resembles a parabola, and the x-axis is the tangent line to this graph at x=0. Check out the other four polynomials.

If you could graph each one on your calculator, that would save you a lot of time.

If you're familiar with calculus, try finding the derivative (dy/dx) and setting that = to 0. That will tell you at which x the graph has a horizontal tangent line which happens to be the x-axis.