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Consider the function $f(x) = x^2 + 3x - 4$.
    \begin{enumerate}
        \item Factor this polynomial and use that to sketch a graph of the function.
        \item This function is not linear, so the `slope' of it changes as we change the $x$ value. We will investigate that here. Find the slope of the line that connects the two points on the graph of $f$ that lie at $x_1 = 1$ and $x_2 = 4$. (Hint: What are the $x$ and $y$ coordinates of these points?)
        \item We are going to define a new function $g_1(x)$ which finds the slope of the line between points on the graph of $f$ above $1$ and $x$. Thus, you found $g_1(4)$ in the last step. Find $g_1(2)$, $g_1(1.5)$, $g_1(1.1)$ and $g_1(1.01)$.
        \item It doesn't make sense to evaluate $g_1(1)$. Why not? If you had to make a guess for $g_1(1)$ based on your previous calculations, what would it be?
        \item Write out a piecewise definition for $g_1(x)$.
        \item Figure out the equation of the line through $(1, f(1))$ with slope $g_1(1)$. Draw this along with a graph of $f$. You may want to use a calculator for this part. What do you notice about how the line relates to the parabola?
        \item Say I was going to have you do all of the previous steps again, but with $x_1 = 3$. (You don't actually have to do this.) Do you think the value of $g_3(3)$ would be larger or smaller than $g_1(1)$? Why? What could you say about the value of $g_{-4}(-4)$? 
    \end{enumerate}

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