91Ó°ÊÓ

The Leading Coefficient Test is a useful tool in graphing polynomials. It helps us determine how the graph behaves at its ends, or as we say in math, the 'end behavior'. When you examine a polynomial, like the quadratic function \( g(x) = -x^2 + 10x - 16 \), the leading coefficient is the number in front of the highest power of \( x \). In this case, it's -1. The degree of the polynomial is the highest power, which here is 2, making it an even polynomial.
For even-degree polynomials, if the leading coefficient is negative, the graph falls as \( x \) approaches both positive and negative infinity. In simpler words, the arms of the graph point downward on both sides. So, for our function \( g(x) \), you can expect that the graph will fall to the left and fall to the right, much like an upside-down "U" shape. This visual guide is crucial as it sets the stage for accurate graph plotting.

Finding the zeros of a polynomial is another critical step. Zeros, also known as roots or solutions, are where the graph crosses or touches the x-axis. For the function \( g(x) = -x^2 + 10x - 16 \), we locate these points by setting the equation to zero. We get:\[0 = -x^2 + 10x - 16\]This can be factored into:\[0 = -(x - 2)(x - 8)\]The solutions are \( x = 2 \) and \( x = 8 \). These x-values are the zeros of the polynomial, meaning they are points on the graph where \( g(x) = 0 \). That means the graph will cross the x-axis at (2,0) and (8,0).
Knowing the zeros gives us anchor points for the graph and helps ensure our sketch will accurately reflect these critical intersections with the x-axis.

After identifying the end behavior via the Leading Coefficient Test and locating the zeros, it's time to bring the graph to life by sketching a continuous curve. This involves plotting several solution points to guide our drawing.
Start with the zeros, \( x = 2 \) and \( x = 8 \). Then select additional points around these zeros. Here, you can plot points such as \( x = 1 \), \( x = 5 \), and \( x = 9 \), and find their corresponding \( y \)-values by substituting back into the function \( g(x) \). These points will plot where the graph hits a particular \( y \)-value for the given \( x \).
With these points sketched, use the information about the falling ends from the Leading Coefficient Test to draw the curve through them in a smooth, continuous line. Remember, the graph should pass through each of these points and reflect the "upside-down U" shape you expect from its end behavior. Continuous curve sketching not only gives you a rough sketch but ensures your graph is reflecting the polynomial's characteristics accurately.