Final answer:
A polynomial function with a negative leading coefficient and even degree has a graph with both ends pointing downwards, and it is symmetric about the y-axis. An example is y = -x² which graphs as a downward opening parabola. Compared to positive and zero slopes, negative slope on a graph shows a decrease in y as x increases.
Step-by-step explanation:
If a polynomial function has a negative leading coefficient and an even degree, the graph of the function generally has a downwards end behavior. That is, both ends of the graph will point downwards towards negative infinity.
As the degree of the polynomial is even, the graph will be symmetric about the y-axis, meaning that the left half of the graph is a mirror image of the right half.
Given that the leading coefficient is negative, if we picture an even-degree polynomial like a parabola (which is a second-degree polynomial) but stretched out to accommodate more complex behavior due to additional terms, it would open downwards.
An example would be the graph of y = -x², an even function where for every positive value of x, there is a negative value of the same magnitude, such that y(x) = −y(-x).
As we observe, when x is positive, the function values decrease, and when x is negative, the same values occur but they are reflections about the y-axis. This demonstrates that the function has a vertical line of symmetry, which is the y-axis itself.
In terms of slopes on the graph, a negative slope indicates that as x increases, y decreases, visible in the downward slant from left to right. This is opposed to a positive slope where y increases with x, and a zero slope indicates a flat, horizontal line on the graph.