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A polynomial function is defined for all x, ranging from minus infinity to plus infinity. Since it is defined for all x, it is defined for x = 0 and this is the point where it intersects the y-axis which is called the y-intercept. It is possible, with suitable choice of coefficients that the function is always positive or always negative. In either case it will not cross the x-axis so that there is no x-intercept. However, it is not true to say that all polynomial functions with real coefficients do not have an x-intercept. In fact all polynomials of odd order (linear, cubic, quintic etc) will have at least one x-intercept.

Q: Why the graph of a polynomial function with real coefficients must have a y intercept but no x intercept?

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no

The zero of a polynomial in the variable x, is a value of x for which the polynomial is zero. It is a value where the graph of the polynomial intersects the x-axis.

A parabola is a graph of a 2nd degree polynomial function. Two graph a parabola, you must factor the polynomial equation and solve for the roots and the vertex. If factoring doesn't work, use the quadratic equation.

This is called the y-intercept and represents the value of the plotted function at x = 0.The place where the graph crosses the y axis is called the y intercept.

That it is non-linear. If it is a graph of a polynomial, it would need to be a polynomial of odd order. But it could be the graph of the tangent function, or a combination of reciprocal functions over a limited domain. In fact the s shaped line, by itself, indicates very little.

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For a polynomial of the form y = p(x) (i.e., some polynomial function of x), having a y-intercept simply means that the polynomial is defined for x = 0 - and a polynomial is defined for any value of "x". As for the x-intercept: from left to right, a polynomial of even degree may come down, not quite reach zero, and then go back up again. A simple example is y = x2 + 1. Why is the situation for "x" and for "y" different? Well, the original equation is a polynomial in "x"; but if you solve for "x", you don't get a polynomial in "y".

no

For a line, this is the x-intercept. For a polynomial, these points are the roots or solutions of the polynomial at which y=0.

The zeros of a polynomial represent the points at which the graph crosses (or touches) the x-axis.

here is the graph

Yes

For a polynomial of order n there are n+1 coefficients that can be changed. There are therefore 2^(n+1) related polynomials with coefficients of the same absolute values. All these generate graphs whose shapes differ.If only the constant coefficient is switched, the graph does not change shape but moves vertically. If every coefficient is switched then the graph is reflected in the horizontal axis. For all other sign changes, there are intermediate changes in the shape of the graph.

A polynomial function have a polynomial graph. ... That's not very helpful is it, but the most common formal definition of a function is that it is its graph. So, I can only describe it. A polynomial graph consists of "bumps", formally called local maxima and minima, and "inflection points", where concavity changes. What's more? They numbers and shape varies a lot for different polynomials. Usually, the poly with higher power will have more "bumps" and inflection points, but it is not a absolute trend. The best way to analyze the graph of a polynomial is through Calculus.

The zero of a polynomial in the variable x, is a value of x for which the polynomial is zero. It is a value where the graph of the polynomial intersects the x-axis.

The y-intercept is the value of the function when 'x' is zero. That is, it's the point at which the graph of the function intercepts (crosses) the y-axis. The x-intercept is the value of 'x' that makes the value of the function zero. That is, it's the point at which 'y' is zero, and the graph of the function intercepts the x-axis.

Depending on the graph, for a quadratic function the salient features are: X- intercept, Y-intercept and the turning point.

Yes.