Linear Functions and Graphs

1. Step 1. Plot the y-intercept. Since b=-1, the y-intercept is (0,-1). So from the y-intercept (or any point), go up 2 units then right 5 units. • • go down 2 units then left 5 units. • This section will focus on linear functions and their applications. A linear function is a function that can be written in the form f(x)=ax+b, where a and b are real numbers. Please note that the variable x is raised to the 1st power (as it is in all linear functions). Also note that x does not (may not) appear in a denominator. Recall that “f(x)” (or g(x), h(x), etc.) is the same as “y”. This means that f(x)=ax+b is the same as y=ax+b, the equation of a straight line with a slope of ‘a’ and a y-intercept of b. Step 2. Use the slope to find two other points. • Step 3. Draw the line through the points.

2. Your Turn Problem # 1 Answer: • • •

3. If a function can be written in the form f(x) = b, where b is a real number, then we call it a constant function. The graph of a constant function is a horizontal line. Example 2. Graph the function f(x) = 5. y axis x axis Next Slide No matter what the x-value is, the corresponding y-value is always 5. Also notice that, although the domain of this function consists of all real numbers, the range is comprised of only one number: 5. The procedure for graphing a constant function is to find the value of b on the y-axis, then draw a horizontal line through that point.

4. Your Turn Problem #2 Answer:

5. Parallel Lines: Two lines are parallel if they have the same slope. Perpendicular Lines: Two lines are perpendicular to each other if their slopes are negative reciprocals of each other. The only exception to the above two methods of determining if two lines are perpendicular are cases involving vertical and horizontal lines. Recall that a horizontal line has a slope of zero, whereas a vertical line has a slope that is undefined. Keep in mind that any horizontal line is perpendicular to any vertical line. In this section, we will not have to concern ourselves with problems involving vertical lines, since vertical lines are not functions. Example 3a. Are the lines f(x)=7x - 4 and g(x)=9x+3 parallel? Solution: The slope of f(x) is 7, whereas the slope of g(x) is 9. Since 7 is not equal to 9, the lines that represent these functions are not parallel. Solution: The slope of g(x) is 9/5 and the slope of h(x) is –5/9. Since 9/5 is the opposite reciprocal of –5/9, the lines are perpendicular. Your Turn Problem #3 Are the lines that represent the following functions perpendicular, parallel, or neither? Answer:

6. Since we have the slope and a point. We can also use the point-slope formula, then write the answer with function notation (y=f(x)). Answer: So, Answer: Your Turn Problem #4 Determine the linear function whose graph is a line with a slope of -4/7 and contains the point (14,9). Example 4. Determine the linear function whose graph is a line that has a slope of 2/3 and contains the point (-1,-4). Solution: We can substitute 2/3 for a in the equation f(x)=ax+b. If the line contains (-1,-4), then we know that when x=-1, y=-4. Another way of writing this information is f(-1)=-4.

7. , Your Turn Problem #5 Determine the linear function whose graph is a line that contains the points (-3,-4) and (1,2). Answer: Example 5. Determine the linear function whose graph is a line that contains the points (3,8) and (7,2). To get an equation of a line, we need one point and the slope. Since we have two points, use the slope formula to find the slope. Now we have a point and the slope. Use the point-slope formula or f(x) = ax +b and simplify.

8. Example 6.Determine the linear function whose graph is a line that contains the point Solution: The slope of the given line is –1/4. The desired line is perpendicular so we want the negative reciprocal which is 4. If the desired line was parallel, we would use the same slope. Now use the point-slope formula, simplify and write in standard form. Your Turn Problem #6

9. Solution: • If we let x = the number of miles driven per day, we can formulate the equation The cost for 1 day driving 300 miles is $88. Your Turn Problem #7 • A retailer has a number of items that she wants to sell and make a profit of 60% of the cost of each item. • Determine a linear function that can be used to calculate selling prices of the items. • Find the selling price on an item that cost $28.60. Answers: a) f(x) = x + 0.6x = 1.6x where x= cost b) The selling price would be $45.76 • Example 7. Rent-A-Wreck Car Rental charges $25 per day plus $0.21 per mile to rent a car. • Determine a linear function whose graph that can be used to calculate daily car rentals. • b) Use the function to determine the cost of renting a car for a day an driving 300 miles. b) let x = 300 to find the total cost for 1 day driving 300 miles. The End B.R. 1-4-07