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ALGEBRA 2: 2.1 Represent Relations and Functions Goal Represent relations and graph linear functions. VOCABULARY Relation A mapping, or pairing, of input values with output values Domain The set of input values in a relation Range The set of output values in a relation Function A relation for which each input has exactly one output Equation in two variables An equation that has an independent or input variable and a dependent or output variable that depends on the value of the input variable Linear function A function that can be written in the form y = mx+ b where m and b are constants REPRESENTING RELATIONS A relation can be represented in the following ways: Ordered Table Graph Mapping Diagram Pairs Input Output (2, 2) x y (2, 2) 2 2 2 2 (0, 1) 2 2 0 2 (3, 1) 3 1 0 1 3 1 Example 1 Identify domain and range. Identify the domain and range of the given relation. Then represent the relation using a graph and a mapping diagram. (5,-2), (-3, -2), (3,3), (-1,-1) Example 2 Identify functions Tell whether each relation is a function. Explain. a. Input Output b. Input Output 2 1 3 6 1 0 4 2 1 2 2 0 2 3 Checkpoint Complete the following exercise. 1. Is the relation given by the ordered pairs (5, 2), (3, 1), (0, 0), (0, 2) and (0, 5) a function? Explain. VERTICAL LINE TEST A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point. Function Not a function Example3 Use the vertical line test Is the relation represented by the graph a function? Explain. a. b. GRAPHING EQUATIONS IN TWO VARIABLES To graph an equation in two variables, follow these steps: Step 1 Construct a table of values. Step 2 Plot enough points from the table to recognize a pattern. Step 3 Connect the points with a line or curve. Example 4 Graph an equation in two variables Graph the equation y = 2x 2. Example 5 Graph the equation y 3 Example 6 Graph the equation x 2 Example 7 Classify and evaluate functions Tell whether the function is linear. Then evaluate the function when x = 3. a. f(x) = 6x + 10 b. g(x) = 2x2 + 4x 1 Checkpoint Complete the following exercises. 2. Use the vertical line test to tell whether the relation is a function. 3. Graph the equation y = 2x 3. Tell whether the function is linear. Then evaluate the function when x = 1. 4. f(x) = 2x3 + 6 x 5. g(x) = 4x + 9 2.2 Find Slope and Rate of Change Goal Find slopes of lines and rates of change. SLOPE OF A LINE Words The slope m of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run). Algebra y y1 m 2 x2 x1 Example 1 Find slope What is the slope of the line passing through the points (1, 3) and (6, 7)? Let (x1, y1) (1, 3) and (x2, y2) (6, 7). CLASSIFICATION OF LINES BY SLOPE The slope of a line indicates whether the line rises from left to right, falls from left to right, is horizontal, or is vertical. Positive slope Rises from left to right Negative slope Falls from left to right Zero slope Horizontal Undefined slope Vertical Example 2 Classify lines using slope Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. a. (6, 2), b. (2, 1), (2, 2) (1, 3) Checkpoint Complete the following exercises. 1. Find the slope of the line passing through the points (4, 2) and (7, -9). 2. Without graphing tell whether the line through the points (3, 4) and (1, 4) rises, falls, is horizontal, or is vertical. SLOPES OF PARALLEL AND PERPENDICULAR LINES Consider two different nonvertical lines l1 and l2 with slopes m1 and m2. Parallel lines The lines are parallel if and only if they have the slope. m1= m2 Perpendicular lines The lines are perpendicular if and only if their slopes are negative reciprocals Example 3 Classify parallel and perpendicular lines Tell whether the lines are parallel or perpendicular. Line 1: through (3, 1) and (2, 5) Line 2: through (3, 4) and (3, 1) Checkpoint Tell whether the lines are parallel, perpendicular, or neither. 3. Line 1: through (1, 0) and (3, 4) Line 2: through (24, 6) and (22, 5) 2.3 Graph Equations of Lines Goal Graph linear equations in slope-intercept or standard form. PARENT FUNCTION FOR LINEAR FUNCTIONS The parent function for the family of all linear functions is y = x. The graph of y = x is shown. In general, a y-intercept of a graph is the y - coordinate of a point where the graph intersects the y-axis. USING SLOPE-INTERCEPT FORM TO GRAPH AN EQUATION Step 1 Write the equation in slope-intercept form by solving for y. Step 2 Identify the y-intercept b and use it to plot the point (0, b) where the line crosses the y -axis. Step 3 Identify the slope m and use it to plot a second point on the line. Step 4 Draw a line through the two points. Example 1 Graph an equation in slope-intercept form 3 Graph y = x + 1. 2 USING STANDARD FORM TO GRAPH AN EQUATION Step 1 Write the equation in standard form. Step 2 Identify the x-intercept by letting y = 0 and solving for x. Use the x-intercept to plot the point where the line crosses the x-axis. Step 3 Identify the y-intercept by letting x = 0 and solving for y. Use the y-intercept to plot the point where the line crosses the y -axis. Step 4 Draw a line through the two points. Example 2 Graph an equation in standard form Graph 2x + 3y = 12. Example 3 Graph horizontal and vertical lines a. Graph y = 1 b. Graph x = 2. Example 4 Graph the equation using any method a. Graph 5 y 10 x 20 b. Graph 4 y 16 0 Checkpoint Graph the equation. 1. y = 2x + 2 4. 5x + 3y = 15 2. y = 4 x 4 3 5. y = 4 3. 4x + 2y = 8 6. 3x = 6 2.4 Write Equations of Lines Goal Write linear equations. WRITING AN EQUATION OF A LINE Use slope-intercept form: Given slope m and y-intercept b, use the equation y = mx + b. Use point-slope form: Given slope m and a point (x1, y1), use the equation y – y1 = m(x- x1 ). Use two points: Given points (x1, y1) and (x2, y2), first use the slope formula to find m. Then use the point-slope form with either given point. Write an equation given the slope and y-intercept Example 1A Example 1B Write an equation of the line shown. Write an equation of a line with m = 2/3 and b = 4 Example 2 Write an equation given the slope and a point Write an equation of the line that passes through (2, 1) and has a slope of 2. Example 3 Write equations of parallel or perpendicular lines Write an equation of the line that passes through (1, 1) and is (a) parallel to, and (b) perpendicular to, the line y = 2x + 3. Write an equation given two points Example 4A Example 4B Write an equation of the line through (3, 1) and (2, 3). Write an equation of the line Example 5 Write an equation in standard form Ax +By = C of the line stat satisfies the given conditions. Use integers for A, B, and C. m = -3/2 and passes through (4, -7) Checkpoint Write an equation of the line in y = mx + b form. 1. 3. Through (2, 3) and (a) parallel and (b) perpendicular to y = 4x 6 2. Through (1, 5) with a slope of 2 4. Through (6, 2) and (3, 2) leave answer in Ax +By = C form 2.7 Use Absolute Value Functions and Transformations Goal Graph and write absolute value functions. PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS The parent function for the family of all absolute value functions is y = | x | . The graph of y = | x | is V-shaped and is symmetric about the y-axis. So, for every point (x, y) on the graph, the point (x, y) is also on the graph. The highest or lowest point on the graph of an absolute value function is called the vertex of an absolute value graph. The vertex of the graph y | x | is (0,0). TRANSFORMATIONS OF GENERAL GRAPHS For | a | 1, the graph is vertically stretched and y = a | x | is narrower than the graph of y = | x |. For | a | < 1, the graph is vertically shrunk and y = a | x | is wider than the graph of y | x |. Example 1 Graph functions of the form y = a | x | Graph (a) y = 1 x and (b) y = 2 | x |. Compare each graph with the graph of y | x |. 3 1 3 Checkpoint Graph the function. Compare the graph with y = | x |. 1. y = 3 | x | Example 2 Graph a function of the form y a | x h | k Graph y = 3 | x 2 | 1. Compare the graph with the graph of y = | x |. Example 3 Write an absolute value equation Write an equation of the graph shown. Checkpoint Complete the following exercises. 1 2. Graph the function y | x 1 | 2. Compare the graph with the graph of 2 y = | x |. 3. Write an equation of the graph shown. 2.8 Graph Linear Inequalities in Two Variables Goal Graph linear inequalities in two variables. Example 1 Checking solutions of inequalities Check whether the ordered pairs (a) (3, 2) and (b) (1, 4) are solutions of 4x + 2y > 6. Checkpoint Check whether the ordered pair is a solution of 2x y 8. 1. (6, 2) GRAPHING A LINEAR INEQUALITY To graph a linear inequality in two variables, follow these steps: Step 1 Graph the boundary line for the inequality. Use a dashed line for < or > and a solid line for or . Step 2 Test a point not on the boundary line to determine whether it is a solution of the inequality. If it is a solution shade the half-plane containing the point. If it is not a solution, shade the other half-plane. Example 2 Graph a linear inequality with one variable Graph y < 1 in a coordinate plane. Example 3 Graph a linear inequality with two variables Graph 3x 2y < 6 in a coordinate plane. Example 4 Graph an absolute value inequality Graph y > 3 x 1+ 2 in a coordinate plane. Checkpoint Graph the inequality in a coordinate plane. 3. x < 2 5. 9x + 3y > 9 4. y x + 2 6. y 2 |x + 2| l