Finding the equation of a polynomial function from its graph involves several steps, typically including analyzing the intercepts, determining the degree, and identifying the leading coefficient.
First, look for the x-intercepts (where the graph crosses the x-axis). Each x-intercept corresponds to a root of the polynomial, which can be used in the function’s equation. For example, if the graph crosses the x-axis at x = 2 and x = -1, these correspond to the factors (x – 2) and (x + 1) respectively.
Next, identify the y-intercept by observing where the graph crosses the y-axis. This point can help you determine the constant term in the polynomial equation.
Then, determine the degree of the polynomial. This can often be inferred from the number of x-intercepts and the behavior of the graph at the extremes. If the graph continues to rise or fall without any upper or lower bounds, the degree is likely odd. Conversely, if it levels off, it may be even.
Once you have the factors from the roots and the y-intercept, you can formulate the polynomial. If, for instance, you identified the x-intercepts as mentioned before and found that the y-intercept is 4, you can set up the polynomial:
f(x) = a(x – 2)(x + 1)
To find ‘a’, plug in the y-intercept:
4 = a(0 – 2)(0 + 1) = -2a
Solving this gives you the value of ‘a’. With ‘a’ known, you’ll end up with the complete polynomial function. This method allows you to derive the polynomial function effectively using the information presented by the graph.