7.1 R eview of Graphs and Slopes of Lines Standard form of a linear equation: The graph of any linear equation in two variables is a straight line. Note: Two points determine a line. Graphing a linear equation: 1.Plot 3 or more points (the third point is used as a check of your calculation) 2.Connect the points with a straight line.

7.1 R eview of Graphs and Slopes of Lines Finding the x-intercept (where the line crosses the x-axis): let y=0 and solve for x Finding the y-intercept (where the line crosses the y-axis): let x=0 and solve for y Note: the intercepts may be used to graph the line.

7.1 R eview of Graphs and Slopes of Lines If y = k, then the graph is a horizontal line (slope = 0): If x = k, then the graph is a vertical line (slope = undefined):

7.1 R eview of Graphs and Slopes of Lines Slope of a line through points (x 1, y 1 ) and (x 2, y 2 ) is: Positive slope – rises from left to right. Negative slope – falls from left to right

7.1 R eview of Graphs and Slopes of Lines Using the slope and a point to graph lines: Graph the line with slope passing through the point (0, 0) Go over 5 (run) and up 3 (rise) to get point (5, 3) and draw a line through both points.

7.1 R eview of Graphs and Slopes of Lines Finding the slope of a line from its equation: 1.Solve the equation for y 2.The slope is given by the coefficient of x Parallel and perpendicular lines: 1.Parallel lines have the same slope 2.Perpendicular lines have slopes that are negative reciprocals of each other

7.1 R eview of Graphs and Slopes of Lines Example: Decide whether the lines are parallel, perpendicular, or neither: 1.solving for y in first equation: 2.solving for y in second equation: 3.The slopes are negative reciprocals of each other so the lines are perpendicular

7.2 Review of Equations of Lines Standard form: Slope-intercept form: (where m = slope and b = y-intercept) Point-slope form: The line with slope m going through point (x 1, y 1 ) has the equation:

7.2 Review of Equations of Lines Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x + 3y = 6 1.solve for y to get slope of line 2.take the negative reciprocal to get the  slope

7.2 Review of Equations of Lines Example (continued): 3.Use the point-slope form with this slope and the point (-4,5) 4.Add 5 to both sides to get in slope intercept form:

7.3 Functions Relations Relation: Set of ordered pairs: Example: R = {(1, 2), (3, 4), (5, 1)} Domain: Set of all possible x-values Range: Set of all possible y-values What is the domain of the relation R?

7.3 Functions Relations Domain: x-values (input) Range: y-values (output) Example: Demand for a product depends on its price. Question: If a price could produce more than one demand would the relation be useful?

7.3 Functions - Determining Whether a Relation or Graph is a Function A relation is a function if: for each x-value there is exactly one y-value –Function: {(1, 1), (3, 9), (5, 25)} –Not a function: {(1, 1), (1, 2), (1, 3)} Vertical Line Test – if any vertical line intersects the graph in more than one point, then the graph does not represent a function

7.3 Functions Function notation: y = f(x) – read “y equals f of x” note: this is not “f times x” Linear function: f(x) = mx + b Example: f(x) = 5x + 3 What is f(2)?

7.3 Functions - Graph of a Function Graph of Does this pass the vertical line test? What is the domain and the range?

7.3 Functions - Graph of a Parabola Vertex

7.4 Variation Types of variation: 1.y varies directly as x: 2.y varies directly as the n th power of x: 3.y varies inversely as x: 4.y varies inversely as the n th power of x:

7.4 Variation Solving a variation problem: 1.Write the variation equation. 2.Substitute the initial values and solve for k. 3.Rewrite the variation equation with the value of k from step 2. 4.Solve the problem using this equation.

7.4 Variation Example: If t varies inversely as s and t = 3 when s = 5, find s when t = 5 1.Give the equation: 2.Solve for k: 3.Plug in k = 15: 4.When t = 5:

9.2 Review – Things to Remember Multiplying/dividing by a negative number reverses the sign of the inequality The inequality y > x is the same as x < y Interval Notation: –Use a square bracket “[“ when the endpoint is included –Use a round parenthesis “(“ when the endpoint is not included –Use round parenthesis for infinity (  )

9.2 Review - Compound Inequalities and Interval Notation Solve each inequality for x: Take the intersection: (why does the order change?) Express in interval notation: 13

9.2 Review - Compound Inequalities and Interval Notation Solve each inequality for x: Take the union: Express in interval notation

9.2 Review - Absolute Value Equations Solving equations of the form:

9.2 Absolute Value Inequalities To solve where k > 0, solve the compound inequality (intersection): To solve where k > 0, solve the compound inequality (union): Why can’t you say ?

9.2 A Picture of What is Happening Graphs of and f(x) = k The part below the line f(x) = k is where The part above the line f(x) = k is where x y f(x) = k

9.2 Absolute Value Inequalities - Form 1 Solving equations of the form: 1.Setup the compound inequality 2.Subtract 4 all the way across 3.Divide by 3 4.Put into interval notation

9.2 Absolute Value Inequalities - Form 2 Solving equations of the form: 1.Setup the compound inequality 2.Subtract 4 all the way across 3.Divide by 3 4.Put into interval notation What part of the real line is missing?

9.2 Absolute Value Inequality that involves rewriting Example: Add 3 to both sides (why?): Set up compound equation: Add 2 all the way across: Put into interval notation

9.2 Absolute Value Inequalities Special case 1 when k < 0: Since absolute value expressions can never be negative, there is no solution to this inequality. In set notation:

9.2 Absolute Value Inequalities Special case 2 when k = 0: Since absolute value expressions can never be negative, there is one solution for this: In set notation: What if the inequality were “<“?

9.2 Absolute Value Inequalities Special case 3: Since absolute value expressions are always greater than or equal to zero, the solution set is all real numbers. In interval notation:

9.2 A Picture of What Happens When k is Negative Graphs of and f(x) = k never gets below the line f(x) = k so there is no solution to and the solution to is all real numbers x y f(x) = k

9.2 Relative Error Absolute value is used to find the relative error of a measurement. If x t represents the expected value of a measurement and x represents the actual measurement, then relative error in

9.2 Example of Relative Error A machine filling quart milk cartons is set for a relative error no greater than.05. In this example, x t = 32 oz. so: Solving this inequality for x gives a range of values for carton size within the relative error specification.

9.2 Solution to the Example 1.Simplify: 2.Change into a compound inequality 3.Subtract 1 4.Multiply by –32 5.Reverse the inequality 6.Put into interval notation