If
f is a function of
x, then the graph of f in the xy-plane consists of all points
(x,f(x)). An
x− intercept is where the graph intersects the
x−axis; since all points on the
x−axis have
y− coordinate 0, the graph of f will cross the x-axis at values of
x such that
f(x)=0. Therefore, the graph of a function
f will have no
x−intercepts if and only if f has no real zeros Likewise, the graph of a quadratic function with no real zeros will have no x-intercepts.
Choice A is incorrect. The graph of a linear function in the xy-plane whose rate of change is not zero is a line with a nonzero slope. The x-axis is a horizontal line and thus has slope 0, so the graph of the linear function whose rate of change is not zero is a line that is not parallel to the
x−axis. Thus, the graph must intersect the x-axis at some point, and this point is an x-intercept of the graph. Choices B and D are incorrect because the graph of any function with a real zeromust have an x-intercept.