Graphing from Factors I

Task

Graph the functions given by the equations $y = (x-1)(x+2)(x-5)$ and $y = 3(x-1)(x+2)(x-5)$ with a viewing window $-10 \le x \le 10$ and $-100 \le y \le 100$. 

1. Describe any similarities you see between the two graphs, and explain how you can see those similarities in the given equations.
2. Write an equation for a function whose graph in the $xy$-plane has $x$-intercepts at -9, -6, 0, and 4. Graph your equation to verify that it works.

IM Commentary

The purpose of this task is to help students understand the relationship between the factors of a polynomial and the $x$-intercepts of the graph of the polynomial. By giving students two different polynomials with the same factors the task draws attention to the fact that both polynomials cross the $x$-axis at the same points. Students are then invited to reflect on why this is so by looking at the structure of the polynomials. In the second part they put their knowledge to use by constructing a polynomial with specified $x$ intercepts.

Solution

1. Both graphs cross the $x$-axis at the same points, $x = -2$, $x = 1$, and $x = 5$. The reason for this is that the expressions on the right hand side of both equations have factors $x+2$, $x-1$, and $x -5$. The first factor is zero when $x = -2$, since $-2+2=0$. The second factor is zero when $x = 1$ and the third factor is zero when $x = 5$. Since the right hand side gives the $y$-coordinate of a point on the graph, the $y$-coordinate is 0 at each of those $x$-values, so the point is on the $x$-axis.
2. We can make a polynomial in $x$ have a zero at any given $x$-value by giving the appropriate factor. So, for example, to make it zero when $x = 4$ we give it the factor $x-4$, and to make it zero when $x = -9$ we give it the factor $x-(-9) = x+9$. A polynomial which is zero at the four specified points is given by  $$y = (x-(-9))(x-(-6))(x-0)(x-4) = (x+9)(x+6)x(x-4).$$