SOLUTION: Explain why the graph of a polynomial function with real coefficients must have a y-intercept but may have no x-intercept.

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Question 495651: Explain why the graph of a polynomial function with real coefficients must have a y-intercept but may have no x-intercept.
Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
The graph could be shift up or down such that it never touches x.
Consider the following graphs:
y = x^2 +1.
graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+x%2Ax%2B1%29
.
y = -x^2 -1.
graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+-x%2Ax-1%29
.
The "why" part of your question is because the function is define for all 'x', from -infinity to +infinity.
So, by definition, it is defined for x=0. That is the y-intercept.
.
As I showed above, you can manipulate the values to avoid the x-axis, at least for even polynomials. But with odd polynomials, including linear equations (the exponent = 1), there will be an x-intercept.
.
Consider:
y = x^3 + 1
graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C+x%5E3%2B1%29