Is it possible for the graph of a polynomial function to have no y-intercept? No x-intercept?
Yes, it is possible for the graph of a polynomial function to have no y-intercept and no x-intercept.
To determine if a polynomial function has a y-intercept, you need to consider its equation. The y-intercept is the point at which the graph intersects the y-axis, where the x-coordinate is always zero (0). Therefore, to find the y-intercept, set x equal to zero (0) in the equation of the polynomial function and solve for y.
If setting x equal to zero results in a valid y-value, the polynomial function has a y-intercept. However, if setting x equal to zero leads to an undefined or non-existent y-value, then the polynomial function does not have a y-intercept.
For example, the equation y = x^2 does have a y-intercept because when x is set to zero, y is also zero. On the other hand, if the equation were y = 1/x, setting x equal to zero would result in a division by zero, which is undefined. Therefore, the graph of y = 1/x does not have a y-intercept.
As for x-intercepts, they are the points where the graph intersects the x-axis, meaning the y-coordinate is zero. To find x-intercepts, you need to set the equation of the polynomial function equal to zero and solve for x.
If solving the equation results in valid x-values, the polynomial function has x-intercepts. However, if solving the equation does not yield any real solutions or leads to complex/imaginary solutions, then the polynomial function does not have x-intercepts.
For example, the equation y = x^2 does have x-intercepts because when y is set to zero, x can be either positive or negative. On the other hand, the equation y = x^2 + 1 does not have x-intercepts because setting y equal to zero leads to no real solutions.
So, in conclusion, a polynomial function can have no y-intercept or x-intercept depending on the equation and the solutions obtained when solving for y = 0 or x = 0.