1. Chapter 1 Functions and Their Graphs
Pre-Calculus
Chapter 1
Functions and Their Graphs

2. 1.1 Lines in the Plane Objectives: Find the slopes of lines.
Write linear equations given points on lines and their slopes.
Use slope-intercept forms of linear equations to sketch lines.
Use slope to identify parallel and perpendicular lines.

3. Warm Up 1.1 – Use “ZOOM SQUARE”
For each set of equations, use your calculator to compare the slopes of the lines.
What do you observe?
Set 1
Set 2
y = 0.5x
y = –0.5x
y = x
y = –x
y = 2x
y = – 2x
y = 4x
y = – 4x

4. Vocabulary Slope of a Line Point-Slope Form of a Line Linear Function
Slope-Intercept Form a Line
General Form of the Equation of a Line
Parallel Lines
Perpendicular Lines
Linear Extrapolation
Linear Interpolation

5. Find the Slope and Equation of the Line
(5, 19)
(10, 6)

6. Definition of the Slope of a Line
Slope = Change in y Change in x

7. Slope of a Line Draw a diagram of each:
A line with positive slope (m > 0) ______________ from left to right.
A line with negative slope (m < 0) ______________ from left to right.
A line with zero slope (m = 0) is ________________.
A line with undefined slope is _________________.

8. Point-Slope Form of Line
Let (x1, y1) be a point on the line whose slope is m.
If (x, y) is any other point on the line, then

9. Point-Slope Equation
The point-slope form of the equation of the line that passes through the point (x1, y1) and has a slope of m is
y – y1 = m (x – x1)
This equation is used frequently in calculus.

10. Example 1: Application Problem
During 2000, Nike’s net sales were $9.0 billion, and in 2001 net sales were $9.5 billion. Write a linear equation giving the net sales y in terms of the year x. Then use the equation to predict net sales for 2002.

11. Slope-Intercept Form
The graph of the equation y = mx + b is a line whose slope is m and whose y-intercept is (0, b).
This can be written as a Linear Function f (x) = mx + b.

12. Example 2
Determine the slope and y-intercept of each linear equation. Then describe the graph of each equation.
3x + 4y = 1
y = 12

13. General Form of Linear Equation
A horizontal line (m = 0) has an equation of the form y = b.
A vertical line has an equation of the form x = a. Can this equation be written in slope-intercept form? Why?
Any line can be written in General Form: Ax + By + C = 0 or Ax + By = c

14. Summary of Equations of Lines
General Form Ax + By + C = 0
Vertical Line x = a
Horizontal Line y = b
Slope-Intercept y = mx + b
Point-Slope y – y1 = m(x – x1)

15. Parallel & Perpendicular Lines
Parallel Lines
Never intersect.
Slopes are equal: m1 = m2
Perpendicular Lines
Intersect at a right angle.
Slopes are negative reciprocals:

16. Example 3
Find the slope-intercept form of the equation of the line that passes through the point (2, –1) and is parallel to the line 2x – 3y = 5.

17. Example 4
Find the slope-intercept form of the equation of the line that passes through the point (2, –1) and is perependicular to the line 2x – 3y = 5.

18. Extrapolation vs. Interpolation
Linear Extrapolation
Use the equation of a line to estimate a point outside the two given points.
Linear Interpolation
Use the equation of a line to estimate a point between the two given points.

19. Additional Example
The number of gallons of gas left in your gas tank can be approximated by a linear function of the number of miles you have driven. Suppose you have driven 123 mi since your last fill-up and you have 13 gal left in your tank. Your car uses 0.05 gal/mi (or 20 mi/gal).

20. Additional Example (cont.)
Write an equation in point-slope form relating the number of miles driven since you filled the tank and the number of gallons left in the tank. Rewrite the equation to express gallons as a function of miles.
Use the answer found in part “a” to determine how many gallons your car’s tank holds.
You want to fill up before the amount left drops below 1 gallon. How much farther can you drive before you must fill up again?

21. Homework 1.1
Worksheet 1.1
# 1, 19, 25, 29, 35, 43, 51, 55, 59, 65, 68, 71, 75 – 78 (matching), 83, 95, 96, 97