Linear Relations and Functions B-3Slope

ACT WARM-UP Simplify 2(6x + 7) − 5(x + 3) Simplify 2(6x + 7) − 5(x + 3) A) 7x − 1B) 7x + 1C) 7x + 19 D) 17x − 1E) 17x + 19 A) 7x − 1B) 7x + 1C) 7x + 19 D) 17x − 1E) 17x + 19 Using the distributive property: 12x + 14 − 5x − 15. Using the distributive property: 12x + 14 − 5x − 15. Combine like terms: (12x − 5x) + (14 − 15). Combine like terms: (12x − 5x) + (14 − 15). Answer is A) 7x − 1 Answer is A) 7x − 1

Objectives Find and use the slope of a line Find and use the slope of a line Graph parallel and perpendicular lines Graph parallel and perpendicular lines

Essential Question How do you find the rate of change? What does the rate of change tell you?

The slope of a linear line is the same, no matter what two points on the line are used.

The formula for slope is often remembered as rise over run, where the rise is the difference in y-coordinates (positive is up, negative is down) and the run is the difference in x-coordinates (positive is right, negative is left). In the field of mathematics the Greek symbol Δ “delta” is used to represent the change in coordinates.

Example 3-1a Find the slope of the line that passes through (1, 3) and (–2, –3). Then graph the line. Slope formula and Simplify.

Example 3-1b Graph the two ordered pairs and draw the line. (1, 3) Use the slope to check your graph by selecting any point on the line. Then go up 2 units and right 1 unit or go down 2 units and left 1 unit. This point should also be on the line. Answer:The slope of the line is 2. (–2, –3)

Example 3-1c Find the slope of the line that passes through (2, 3) and (–1, 5). Then graph the line. Answer:The slope of the line is

Example 3-2a Graph the line passing through (1, –3) with a slope of Graph the ordered pair (1, –3). Then, according to the slope, go down 3 units and right 4 units. Plot the new point at (5, –6). (1, –3) (5, –6) Draw the line containing the points.

Example 3-2b Graph the line passing through (2, 5) with a slope of –3. Answer:

The slope of a line tells the direction in which it rises or falls. If the line rises to the right, than the slope is positive. If the line is horizontal, then the slope is zero. If the line falls to the right, then the slope is negative. If the line is vertical, then the slope is undefined... (2,4) (-5,-3).. (-3,4)(3,4).. (-2,4) (6.-3).. (3,3) (3,-3)

Slope is often referred to as rate of change. It measures how much a quantity changes, on average, relative to the change in another quantity, often time. The rate of change is constant. Functions with a constant rate of change are called linear functions.

Example 3-3a Communication Refer to the graph. Find the rate of change of the number of radio stations on the air in the United States from 1990 to Slope formula Substitute.

Example 3-3b Simplify. Answer: Between 1990 and 1998, the number of radio stations on the air in the United States increased at an average rate of 0.225(1000) or 225 stations per year.

Example 3-3c Computers Refer to the graph. Find the rate of change of the number of households with computers in the United States from 1984 to Answer: The rate of change is 2.9 million households per year.

A family of graphs is a group of graphs that displays one or more similar characteristics. The parent graph is the simplest of the graphs in a family. It is often the one that passes through the origin, where the values of x and y are both zero. The parent graph of a linear function is f(x) = x.

Parallel Lines In a plane, nonvertical lines with the same slope are parallel. All vertical lines are parallel with an undefined slope. All horizontal lines are parallel because they all have a slope of 0.

Example 3-4a Graph the line through (1, –2) that is parallel to the line with the equation The x -intercept is –2 and the y -intercept is 2. Use the intercepts to graph The line rises 1 unit for every 1 unit it moves to the right, so the slope is 1. Now, use the slope and the point at (1, –2) to graph the line parallel to (1, –2)(2, –1)

Example 3-4b Graph the line through (2, 3) that is parallel to the line with the equation Answer:

Perpendicular Lines Perpendicular lines have slopes that are opposite reciprocals of each other. This relationship is true in general. In a plane, two oblique lines are perpendicular if and only if the product of their slopes is – 1. An oblique line is a line that is neither horizontal nor vertical. Any vertical line is perpendicular to any horizontal line.

Example 3-5a Graph the line through (2, 1) that is perpendicular to the line with the equation The x -intercept is or 1.5 and the y -intercept is –1. Use the intercepts to graph 2x – 3y = 3 The line rises 1 unit for every 1.5 units it moves to the right, so the slope is or

2x – 3y = 3 Example 3-5b Graph the line through (2, 1) that is perpendicular to the line with the equation The slope of the line perpendicular is the opposite reciprocal of or Start at (2, 1) and go down 3 units and right 2 units. (2, 1) (4, –2) Use this point and (2, 1) to graph the line.

Example 3-5c Graph the line through (–3, 1) that is perpendicular to the line with the equation Answer:

Essential Question How do you find the rate of change? Use the slope formula What does the rate of change tell you? It measures how much a quantity changes, on average, relative to the change in another quantity, often time.

Math Humor Why was the student afraid of the y-intercept? Why was the student afraid of the y-intercept? She thought she’d be stung by the b. She thought she’d be stung by the b.