Introductory Calculus: Factoring, Slope, and Graphing

FACTORING: Factoring is the “undoing” of multiplication, putting an expression into terms which are multiplied. For example, we know that 6x(x³ – 7) = 6x⁴ – 42x.

The expression 6x⁴ – 42x factors into two terms which are multiplied: 6x•(x³ – 7)

The difference of two squares such as x²- 49 factors into (x+7)•(x-7)

Sometimes trinomials factor very nicely:

(2x² - x - 6) = (2x + 3)•(x - 2)

Sometimes they do not. The trinomial x² + x + 1 defies a simple factoring.

Examples:

x² - a²

4x² - 9y²

x⁴ - 16

x² + 4

x² - 5x + 6

a² + 8a - 20

x² + 3x - 5

3x² - 13x - 10

SLOPE: The slope of a line connecting two points is a ratio of the “rise” to the “run,” which is a ratio of the vertical distance between the points to the horizontal distance between the two points.

A line passing through the points (2, 5) and (– 3, 1) has a slope of

Since this is a positive number, the line will appear to slope upwards to the right when graphed.

Examples:

What is the slope of the line through points (1, –2) and (3, 4)?

What is the slope of the line through points

and

A line has slope 4. It passes (1, 3) and (x, 7) what is x?

A line has slope ¾. It passes through points (5, y) and (4, -3). What is y?

GRAPHING: A graph is a more visual representation of the points described by a rule, list, or expression.

We will use the TI 83/84 family calculator extensively during this course. To graph a line with slope –2 and y–intercept (0, 3) we note that the equation for this line is y=-2x + 3 and enter this as Y1 in the calculator. It is actually faster to sketch this by hand without the calculator. The Y= screen, the WINDOW, and the graph are shown below.

Some functions are defined in separate pieces and are called “piece-wise defined” functions. If the line shown above is only valid for x > –1 and the remainder of the function is defined by y = 4 - x² we have the following definition and graph of this function:

Examples:

Graph the following piecewise-defined function on the axes below. Label accurately and completely.