Building an Explicit Quadratic Function by Composition

Task

Let $f$ be the function defined by $f(x) = 2x^2 + 4x - 16$. Let $g$ be the function defined by

1. Verify that $f(x) = g(x)$ for all $x$.
2. In what ways do the equivalent expressions $2x^2 + 4x - 16$ and $ 2(x+1)^2 - 18$ help to understand the function $f$?
3. Consider the functions $h,l,m,$ and $n$ given by
   \begin{eqnarray} h(x) &=& x^2 \\ l(x) &=& x + 1 \\ m(x) &=& x - 9 \\ n(x) &=& 2x \end{eqnarray}
   Show that $f(x)$ is a composition, in some order, of the functions $h,l,m,$ and $n$. How do you determine the order of composition?
4. Explain the impact each of the functions $l,m,$ and $n$ has on the graph of the composition.

IM Commentary

This task is intended for instruction and to motivate the task Building a General Quadratic Function. There are often many equivalent expressions which give rise to the same function: $2x^2 + 4x - 16$ and $2(x+1)^2 -9$ are the two examples studied in this problem. Often times, the choice of one particular expression is motivated by what it reveals about the underlying function. In this case, for example, the expression $2x^2 + 4x - 16$ allows us to conclude right away that the function is quadratic and that its graph is a parabola which opens upward. The expression $2(x+1)^2 -18$, on the other hand, helps us find the $x$-coordinate of the vertex of the parabola quickly and also to find the two values of $x$, $2$ and $-4$, which make the expression equal to zero.

This task assumes that the students are familiar with the process of completing the square. This is vital as otherwise the expression $2(x+1)^2 - 18$ will appear to come from nowhere. To briefly recall the process, in a series of equations, we have

The first step here, factoring out a $2$, definitely corresponds to a multiplicative scaling, but for the rest of the algebra, consisting in ''completing the square,'' more work is necessary to connect this to the transformational approach taken in this problem.

It is also important to note that although the transformations are the focus of part (d) of this problem, the rest of the problem has a strongly algebraic flavor. This is in order to connect the transformations with a method of deriving the quadratic formula which is accomplished in ''Building a general quadratic function.'' For more practice with the geometric aspects of the transformations, the tasks ''Building a quadratic function from $f(x) = x^2$'' and ''Transforming the graph of a quadratic function'' directly address this.

Finally, note that the task depends on composition of functions, which is included in the (+) standard F-BF.1.c. It would need to be reformulated for students in courses that do not include the (+) standards.

Solution

1. To verify that $f$ and $g$ are the same function, we begin by calculating $(x+1)^2 = x^2 + 2x + 1$. So $(x+1)^2 - 9 = x^2 + 2x - 8$ and finally, multiplying both sides of this equation by $2$, $$ 2(x+1)^2 - 18 = 2x^2 + 4x - 16. $$ Thus $f(x)= g(x)$ for all $x$ and so $f$ and $g$ are the same function.

2. The expression $2x^2 + 4x -16$ for the value of $f$ at $x$ makes it clear that the graph of $f$ is a parabola and that this parabola opens upward. This expression also allows us to see readily where this graph crosses the $y$-axis as when $x= 0$ we see that $f(x) = -16$. The expression $2(x+1)^2 - 18$, on the other hand, allows us to readily find the $x$-coordinate of the vertex of the graph: indeed this expression has minimal value when $(x+1) = 0$ or when $x = -1$. In addition, this expression for $f$ allows us to find the $x$-intercepts of the graph of $f$, that is the values of $x$ for which $f(x)= 0$. It is sufficient, for this purpose, to solve the equation $2(x+1)^2 - 18= 0$ which we may do by dividing both sides by two, adding $9$ to both sides, taking the square root of both sides, and finally subtracting one from each side. The $x$-intercepts are $x = -4$ and $x = 2$.

3. Rewriting the expression for $f(x)$ $$ 2(x+1)^2 - 18 = 2\left((x+1)^2 - 9\right) $$ we can describe in words how this function treats an input $x_0$: first add $1$, then square the result, then subtract $9$, and finally, multiply by $2$. If we look at the four functions $h,l,m,n$ the one which adds one is $l$, the one which squares the input is $h$, the one which subtracts $9$ is $m$, and the one which multiplies by $2$ is $n$. Keeping the order in mind, this shows that $$ f(x) = 2\left((x+1)^2 - 9 \right) = n(m(h(l (x)))) $$ for all real numbers $x$.

   The order in which the functions are composed is definitely important. For example $h( l(x)) = (x+1)^2$ while $l(h(x)) = x^2 + 1$ and these are not the same function as we see, for example, by plugging in $x = 2$. An efficient way to determine the order of the composition is what we have done, namely to formulate in words what $f$ does with a given input and this will tell us the order of the composition.

4. The impact of $l$, replacing $x$ by $x + 1$ is a horizontal shift: the graph shifts to the left by $1$ unit. The impact of $h$ is to change its linear input of $x+1$ to a quadratic output of $(x+1)^2$. The impact of $m$, subtracting $9$ from its input, is a vertical shift: the graph is moved downward by $9$ units. Finally the impact of $n$, multiplying by $2$, is to scale the graph, making it more steep but preserving its orientation (that is the graph opens upward both before and after $n$ is applied).

   It is important to note here that the functions $l$ and $m$ look very similar. They are both linear functions and the leading coefficient in both cases is $1$. In both cases, these induce ''shifts'' in the graph of $h(x) = x^2$. The reason why $l$ induces a horizontal shift while $m$ gives a vertical shift is that $l$ is performed before the squaring function $h$ while $m$ is performed after the squaring function $h$. In other words, whether or not a linear function acts as a vertical or horizontal shift depends on its place in the composition of functions.