Problem 87 - Algebra and Trigonometry

Question

Can the graph of a polynomial function have no \(y\) -intercept? Explain.

Step-by-Step Solution

Answer

No, the graph of a polynomial function will always have a y-intercept because the y-intercept is the constant term of the function.

Step 1: Understanding a polynomial function

A polynomial function is an equation made up of variables and coefficients where variables are only ever raised to a positive, integer power. The general form of a polynomial function of degree n is: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where the \(a_i\) are constants and \(a_0\) is the y-intercept.

Step 2: Understanding the y-intercept

The y-intercept of a function is the value of \(y\) when \(x=0\). For a polynomial function, setting \(x=0\), we get \(y=a_0\). So, the y-intercept of a polynomial function is \(a_0\), which is a constant term of the function.

Step 3: Determine if a polynomial function can have no y-intercept

As defined in Step 2, the y-intercept of a polynomial function is the constant term. In other words, when \(x=0\), \(y=a_0\). Unless \(a_0\) is undefined or not a real number, \(y=a_0\) will be a point on the y-axis. Therefore, the graph of a polynomial function will always have a y-intercept.