1.3 Linear Functions Linear Function A function f defined by where a and b are real numbers, is called a linear function. Its graph is called a line. Its solution is an ordered pair, (x,y), that makes the equation true.

Example 1.3 Linear Functions The points (0,6) and (–1,3) are solutions of since 6 = 3(0) + 6 and 3 = 3(–1) + 6.

1.3 Graphing a Line Using Points Graphing the line Connect with a straight line. x y 2 1 3 6 1 9 (0,6) and (–2,0) are the y- and x-intercepts of the line y = 3x + 6, and x = –2 is the zero of the function.

1.3 Graphing a Line with the TI-83 Graph the line with the TI-83 Xmin=-10, Xmax=10, Xscl=1 Ymin=-10, Ymax=10,Yscl=1

Locating x- and y-Intercepts To find the x-intercept of the graph of y = ax + b, let y = 0 and solve for x. To find the y-intercept of the graph of y = ax + b, let x = 0 and solve for y.

1.3 Zero of a Function Zero of a Function Let f be a function. Then any number c for which f (c) = 0 is called a zero of the function f.

1.3 Graphing a Line Using the Intercepts Example: Graph the line . x y x-intercept 5 y-intercept 2.5

1.3 Application of Linear Functions A 100 gallon tank full of water is being drained at a rate of 5 gallons per minute. a) Write a formula for a linear function f that models the number of gallons of water in the tank after x minutes. b) How much water is in the tank after 4 minutes? c) Use the x- and y-intercepts to graph f. Interpret each intercept.

1.3 Constant Function Constant Function b is a real number. The graph is a horizontal line. y-intercept: (0,b) Domain range Example:

1.3 Constant Function Constant Function A function defined by f(x) = b, where b is a real number, is called a constant function. Its graph is a horizontal line with y-intercept b. For b not equal to 0, it has no x-intercept. (Every constant function is also linear.)

1.3 Graphing with the TI-83 Different views with the TI-83 Comprehensive graph shows all intercepts

1.3 Slope Slope of a Line In 1984, the average annual cost for tuition and fees at private four-year colleges was $5991. By 2004, this cost had increased to $20,082. The line graphed to the right is actually somewhat misleading, since it indicates that the increase in cost was the same from year to year. The average yearly cost was $705.

1.3 Formula for Slope The slope m of the line passing through the points (x1, y1) and (x2, y2) is

1.3 Example: Finding Slope Given Points Determine the slope of a line passing through points (2, 1) and (5, 3).

1.3 Graph a Line Using Slope and a Point Example using the slope and a point to graph a line Graph the line that passes through (2,1) with slope

1.3 Slope of a Line Geometric Orientation Base on Slope For a line with slope m, If m > 0, the line rises from left to right. If m < 0, the line falls from left to right. If m = 0, the line is horizontal.

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

1.3 Vertical Line Vertical Line A vertical line with x-intercept a has an equation of the form x = a. Its slope is undefined.

1.3 Slope-Intercept Form of a Line The slope-intercept form of the equation of a line is where m is the slope and b is the y-intercept.

1.3 Matching Examples Solution: A. B. C. 1) C, 2) A, 3)B

Interpreting Slope 1.3 Application of Slope In 1980, passengers traveled a total of 4.5 billion miles on Amtrak, and in 2007 they traveled 5.8 billion miles. a) Find the slope m of the line passing through the points (1980, 4.5) and (2007, 5.8). Solution: b) Interpret the slope. The average number of miles traveled on Amtrak increased by about 0.05 billion, or 50 million miles per year.