Graph, Equations and Inequalities Chapter 2 Graph, Equations and Inequalities

2.1 GRAPHS Cartesian coordinate system Origin x-axis, y-axis ordered pair, x- coordinate, y- coordinate Quadrants Equation Solution of an equation in 2 variables Graph of an equation x-intercept, y- intercept

2.2 EQUATIONS OF LINES Given 2 points (x1, y1), (x2, y2) x (change in x) = x2 – x1 y (change in y) = y2 – y1 Slope of a line: The slope of the line though the two points (x1, y1) and (x2 , y2), where (x1x2) is defined as the quotient of the change in y and the change in x, or

Slope and intercepts The slope of every horizontal line is 0. The slope of every vertical line is undefined. If k a constant, then the graph of the equation y = k is the horizontal line with y-intercep k. If k is a constant, then the graph of the equation x = k is vertical line with x- intercept k.

Slopes of Parallel and Perpendicular lines Two nonvertical lines are parallel whenever they have same slope. Two nonvertical lines are perpendicular whenever the product of their slopes is –1.

SLOPE – INTERCEPT FORM If a line has slope m and y-intercept b, then it is the graph of the equation y = mx + b. This equation is called the slope- intercept form of the equation of the line.

POINT – SLOPE FORM If a line has slope m and passes though the point (x1 , y1), then y – y1 = m(x – x1 ) is the point-slope form of equation of the line.

Linear Equations Equation Description ax + by = c x = k y = k y = mx+b y -y1 = m(x-x1) General form. If a  0 and b  0, the line has x-intercept c/a and y-intercept c/b. Vertical line, x-intercept k, no y-intercept, undefined slope. Horizontal line, y-intercept k, no x-intercept slope 0 Slope-intercept form, slope m, y-intercept b Point-slope form, slope m, the line passes thought (x1 , y1).

APPLICATIONS OF LINEAR EQUATIONS Example 1: Celsius and Fahrenheit temperature. 32F corresponds to 0C, 212F corresponds to 100C. The relationship between Celsius and Fahrenheit temperature is linear. Write the equation of this relationship. Find the Celsius temperature corresponding to 75F.

APPLICATIONS OF LINEAR EQUATIONS Example 2: Break-even point. Profit = Revenue – Cost p = r – c Cost of producing x units: c = 20x + 100 The unit price is $24/unit: r = 24x Find the break-even point. Graph of cost and revenue equation.

2.4 LINEAR INEQUALITIES Inequality: statement that one mathematical expresion is greater than (or less than) another. Linear inequality Example: 4 – 3x ≤ 7 + 2x

Properties of Inequalities For real numbers a, b, c: If a < b, then a + c < b + c If a < b and c > 0, then ac < bc If a < b and c < 0, then ac > bc Note: The properties are valid not only for <, but also for >, ≤, ≥. Pay attention to 3) with the change of direction of the inequality symbol.

Examples Solve 3x + 5 > 11 Solve 4 – 3x ≤ 7 + 2x Solve – 2 < 5 + 3m < 20 The formula for converting from Celsius to Fahrenheit is: F = (9/5)C + 32 What Celsius range corresponds to the range from 32ºF to 77ºF?

Inequalities with absolute values Recall: The absolute value of a number a is the distance from 0 to a on the number line. Assume a and b are real numbers with b positive. Solve |a| < b by solving –b < a < b Solve |a| > b y solving a < –b or a > b

Examples Solve |x – 2| < 5 Solve |2 – 7m| – 1 > 4 Write statement using absolute value: k is at least 4 units from 1. Write statement using absolute value: p is within 2 units of 5.