1. Some "Special" Functions f(x) = |x|, the absolute value function. The domain is the set of all real numbers. The graph is symmetric with respect to the y- axis. The range of f is the set of nonnegative real numbers. The x-intercept and y-intercept are both 0. Graph of y = |x|

2. More "Special" Functions f(x) = x 2, the squaring function. The domain is the set of all real numbers. The graph is symmetric with respect to the y- axis. The range of f is the set of nonnegative real numbers. The x-intercept and y-intercept are both 0. Graph of y = x 2

3. More "Special" Functions f(x) =, the square root function. Since is not defined for x < 0, the domain is the set of nonnegative real numbers. The range of f is the set of nonnegative real numbers. The x-intercept and y-intercept are both 0. Graph of y =

4. More "Special" Functions f(x) = x 3, the cubic function. The domain is the set of all real numbers. The graph is symmetric with respect to the origin. The range of f is the set of all real numbers. The x-intercept and y-intercept are both 0. Graph of y = x 3

5. Graphing Techniques--Vertical Shift If p > 0, the graph of y = f(x) + p is the graph of y = f(x) shifted up p units. Similarly, the graph of y = f(x) – p is the graph of y = f(x) shifted down p units. y = x 2 + 1 y = x 2 y = x 2 – 0.5

6. Graphing Techniques--Horizontal Shift If p > 0, the graph of y = f(x – p) is the graph of y = f(x) shifted p units to the right. Similarly, the graph of y = f(x + p) is the graph of y = f(x) shifted p units to the left. y = x 3 y = (x–2) 3 y = (x+2) 3

7. Graphing Techniques--Reflection The graph of y = –f(x) is the reflection about the x-axis of the graph of y = f(x). The graph of y = f(–x) is the reflection about the y-axis of the graph of the graph of y = f(x). Reflection of y=x 2 about x-axis Reflection of y=|x–1| about y-axis

8. Graphing Techniques--Stretching and Shrinking If k > 1, the graph of y =k f(x) is the graph of y = f(x) stretched vertically by k units. If 0 < k < 1, the graph of y = kf(x) is the graph of y = f(x) shrunk vertically by k units. y=2x 2 y=x 2 y=0.5x 2

9. Piecewise Defined Functions When a function is defined in different ways over different parts of its domain, it is said to be a piecewise-defined function. Example. The price of a movie ticket changes from $5 to $8 at 5pm. We can write the cost C(t) of a ticket, in dollars, as the following piecewise-defined function of time t. We recall that the absolute value function is also a piecewise- defined function.

10. Increasing and Decreasing Functions If the graph of function f steadily rises as we go from left to right, we say that f is increasing. Similarly, if the graph of function f steadily falls as we go from left to right, we say that f is decreasing. In general, a function may increase over some intervals, decrease over others, and remain constant in still other intervals. increasing decreasing

11. Linear and Quadratic Functions The polynomial function is called a linear function. Its graph is a straight line. The polynomial function of second degree is called a quadratic function. Its graph is in the shape of a parabola. For example, y = 2x 2 – 4x + 3

12. Summary of Graphs of Functions; We discussed Graphs of the absolute value function, the squaring function, the square root function, and the cubic function Vertical shift and horizontal shift Reflection about the x-axis and about the y-axis Vertical stretching and shrinking Piecewise defined functions Increasing and decreasing functions Linear and quadratic functions

13. Linear Functions The slope m of a line that is not vertical is given by where on the line. Example. Find the slope of the line that passes through (5, 6) and (1, –2). Slope measures the steepness of a line. On the line in the previous example, for every increase of 1 in x, we get an increase of 2 in y.

14. Properties of slope Let m be the slope of a line. 1. When m > 0, the line is the graph of an increasing function. 2. When m < 0, the line is the graph of a decreasing function. 3. When m = 0, the line is the graph of a constant function. 4. Slope does not exist for a vertical line, and a vertical line is not the graph of a function. m = 3 m = 1 m = 1/2 m = –3 m = –1 m = –1/2

15. Equation of a line: Point-Slope Form The equation is that of a line with slope m that passes through the point (x 1, y 1 ). Example. Find an equation of the line that passes through the points (6, –2) and (–4, 3). First we calculate the slope m = –0.5. We let (x 1, y 1 ) = (6, –2), although we could have validly used the other point. The equation is:

16. Equation of a line: Slope-Intercept Form The equation is that of a line with slope m and y-intercept b. Example. Find an equation of the line that passes through the points (1, 2) and (0, 3). We calculate the slope as m = –1. Since we are given that the y-intercept = 3, the equation is:

17. Horizontal and Vertical Lines The equation is that of the horizontal line through the point (a, b). The slope of a horizontal line is 0. The equation is that of the vertical line through the point (a, b). A vertical line has undefined slope.

18. General First Degree Equation of a Line The graph of the general first-degree equation is a line. If A = 0, the graph is a horizontal line. If B = 0, the graph is a vertical line. Example. Convert x + 2y + 3 = 0 to slope-intercept form.

19. Parallel and Perpendicular Lines Two lines with slopes m 1 and m 2 are parallel if and only if Two lines with slopes m 1 and m 2 are perpendicular if and only if Example. Given the line y = 3x – 2, find an equation of the line through (–5, 4) that is (a) parallel to the given line, (b) perpendicular to the given line (a) (b)

20. Linear Functions; We discussed Slope of a line Slope for increasing, decreasing, and constant linear functions Point-slope form Slope-intercept form Horizontal and vertical lines General first degree equation Parallel and perpendicular lines