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Chapter 3: Problem 87

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

Short Answer

Expert verified
No, the graph of a polynomial function always has a y-intercept because when substituting \(x = 0\) in to the function, we get a constant, \(a_0\), which represents the point on the y-axis where the function intersects or touches.

Step by step solution

01

Define Polynomial Function and Y-intercept

A polynomial function is a function that can be expressed in the form \(f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0\), where \(a_n, a_{n-1}...\) are constants and \(x\) is a variable. A y-intercept is a point where the graph of the function intersects the y-axis. This occurs when \(x=0\).
02

Determine if Polynomial Function can have no Y-intercept

Substitute \(x=0\) in the polynomial function. You obtain \(f(0)=a_n*0^n+a_{n-1}*0^{n-1}+...+a_2*0^2+a_1*0+a_0\), which simplifies to \(f(0) = a_0\). This means, every time we input \(0\) for \(x\) in the polynomial function, we get \(a_0\) for \(f(x)\), which is a constant. This shows that the polynomial function always intersects or touches the y-axis at the point \(y=a_0\). Therefore, it is impossible for a polynomial function to have no y-intercept.

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