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

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.

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

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

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

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

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 _________________.

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

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.

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.

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.

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

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

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)

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

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.

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.

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.

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).

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?

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