Question

Explain the relationship between the factors of a quadratic expression, the roots of the related quadratic equation, and the x-intercepts of the graph of the related function.

Answer 1

The zeros of the function are the numbers that when you plug them into the definition of the function the answer is zero. In other words, the number 'a' is a zero of the function f, if f(a)=0.

The x-intercepts of f are the points on the x-axis where the graph of f crosses the x-axis.

So the intercepts are points, i.e. geometric things, but the zeros are numbers, not geometric things.

However, if you identify the points on the x-axis with numbers, that is, think of the points as being real numbers, then the zero 'a' of f is becomes the x-intercept, x=a on the number line. That is, once you blur the distinction between points and numbers, the two concepts, zeros and x-intercepts become "the same".

This kind of blurring of different things is common in mathematics. For example a function is technically a set of ordered pairs, but usually one thinks of a graph on the x-y plane or a relationship between two quantities, rather than a set.

Rational numbers are equivalence classes of ordered pairs of integers. But no one thinks of that when looking at the fraction 3/4.