Consider the function f(x) = x2 + 3x - 4. (a) Factor this polynomial and also complete the square. Draw a graph of the function. Label the roots and vertex. (b) This function is not linear, so the "slope" changes as we change the x value. We will investigate that here. Find the slope of the line that connects the points (1, f(1)) and (4, f(4)) on the graph of f. (c) We define a new function gı (2) which finds the slope of the line between the points (1, f(1)) and (x, f(x)) on the graph of f. In the last step you calculated 91(4). Find 91(2), 91(1.5), 91(1.1), and 91(1.01). (d) Explain why (1) is undefined. Based on the values you calculated in part (c). what do you think gı(1) should be? Use this value for 91(1) from now on. (e) Find the equation of the line through (1, f(1)) with slope 91 (2) and draw it along with the graph of f. What do you notice about how the line relates to the parabola? (f) Define 93 (2) to be the slope of the line through (3, f(3)) and (x, f(x)). Again 93(3) is undefined. Imagine going through the steps above to find a logical value for 93(3). (You do not actually have to do this.) Do you think the value of q:(3) would be larger or smaller than 91(1)? Why? What about 9-5(-5)?
Consider the function f(x) = x2 + 3x - 4. (a) Factor this polynomial and also complete the square. Draw a graph of the function. Label the roots and vertex. (b) This function is not linear, so the "slope" changes as we change the x value. We will investigate that here. Find the slope of the line that connects the points (1, f(1)) and (4, f(4)) on the graph of f. (c) We define a new function gı (2) which finds the slope of the line between the points (1, f(1)) and (x, f(x)) on the graph of f. In the last step you calculated 91(4). Find 91(2), 91(1.5), 91(1.1), and 91(1.01). (d) Explain why (1) is undefined. Based on the values you calculated in part (c). what do you think gı(1) should be? Use this value for 91(1) from now on. (e) Find the equation of the line through (1, f(1)) with slope 91 (2) and draw it along with the graph of f. What do you notice about how the line relates to the parabola? (f) Define 93 (2) to be the slope of the line through (3, f(3)) and (x, f(x)). Again 93(3) is undefined. Imagine going through the steps above to find a logical value for 93(3). (You do not actually have to do this.) Do you think the value of q:(3) would be larger or smaller than 91(1)? Why? What about 9-5(-5)?
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