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Lesson 1.7 Linear Functions

Lesson 1.7 Linear Functions. When there is a constant increase or decrease in one variable with respect to another, a linear relationship occurs between the variables. y. b. a. x. d. c. A function f defined by a linear equation of the form y = f(x) = ax + b, where

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Lesson 1.7 Linear Functions

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  1. Lesson 1.7 Linear Functions

  2. When there is a constant increase or decrease in one variable with respect to another, a linear relationship occurs between the variables. y b a x d c A function f defined by a linear equation of the form y = f(x) = ax + b, where a and b are constants, is called a linear function. Linear functions have graphs that are straight lines. Note: Vertical lines are not linear functions.

  3. Slope of a line The quotient of the difference of the y-coordinates over the difference of the x-coordinates. ANY TWO DISTINCT POINTS ON A LINE WILL GIVE THE SAME SLOPE Point Slope equation of a line. The line with slope m that passes through the point has the point slope equation

  4. The slope of a line determines its direction. 1. A line with positive slope is directed upward. 2. A line with negative slope is directed downward. 3. A line with zero slope is horizontal. Slope intercept equation of a line The line with slope m passing through the point Has the slope intercept equation y = mx + b, where b =

  5. Ex 1: A line passes through the points (2, 3) and (-4, 0). a.) Find a point-slope equation of this line b.) Find the slope intercept equation of this line Solution: a.) Using any point… (2, 3) gives... If using point (-4, 0)… b.) Both point slope equations reduce to the same slope intercept equation.

  6. Ex 2: In 1960 there were 317 ppm of carbon dioxide in the atmosphere and 361 ppm in 1995. Determine a linear function to predict the level of carbon dioxide in the atmosphere in the year 2020. Solution: Find slope. The line has equation…

  7. Ex 3: Sketch the graph of the linear function described by Since the slope is negative, the line will be decreasing. The y-intercept is (0, 2). There are several ways to graph this line: • Intercepts • t-table • Slope intercept (rise over run and y-intercept) y (0,2) 2 (3,0) x 3

  8. Parallel and Perpendicular lines Straight lines that never intersect are called parallel lines. Straight lines that intersect at right angles are perpendicular lines. Slopes of parallel and perpendicular lines • Suppose the nonvertical lines l and m have the slopes m1 and m2. • Lines l and m are parallel if and only if m1 = m2. • Lines l and m are perpendicular if and only if m1m2 = -1 Ex 4: Find the equation of the line passing through (3, 1) and a. Parallel to the line with equation y = 2x – 1; b. Perpendicular to the line with equation y = 2x – 1. Solution: The slope of the new line must be 2. Use the point slope equation of a line and the new slope. y – 1 = 2(x – 3) y = 2x – 6 + 1 y = 2x - 5

  9. Our new line must have a negative reciprocal slope. The given line • has slope of 2. So our new line has slope -1/2. Use the point slope form of a line to write the equation:

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