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3.2 Graphing Functions and Relations • Another way that we can determine if a relation is a function is by looking at a graph of the relation. • A relation is a function if and only if no vertical line intersects the graph more than once. Graph the relation and then state whether or not the relation is a function. 1. 0, 4 , 3,0, 1,2, 3,1, 4,0 Graph the relation and then state whether or not the relation is a function. 2. 1,1, 1,2, 4,2, 5, 3, 2,3 Graph the relation defined by each of the following rules over the domain 2,1,0,1,2, 3 3. y5 x Graph the relation defined by each of the following rules over the domain 2,1,0,1,2, 3 4. xy3 3.3 Graphing Linear Equations Linear Equations • Linear equations contain both an • The graph of a linear equation is a x and a line y . • There are two forms of linear equations that we will look at: Slope-intercept 1. , which is in the form y mx b 2. Standard , which is in the form Ax By C Slope-Intercept Form y mx b • The m represents the slope of the line. The slope is steepness the of the line. – When the slope is positive, the graph goes up from left to right. – When the slope is negative, the graph goes down from left to right. rise – When graphing, the slope can be thought of as run over , where the numerator determines the displacement and the denominator vertical horizontal determines the displacement between points on the line. Slope-Intercept Form y mx b • The b represents the of the line. y-intercept In other words, it’s where the line crosses the y-axis . Determine the slope and y-intercept of the lines below. 1. 1 y x4 2 2. 3y x 12 Graphing in Slope-Intercept Form • First, make sure that the equation is in the form y=mx+b . • Plot the y-intercept first. Use this as your starting point. • Use the slope to plot another point, using the idea of rise over run. • Draw a line through your points with arrows on each end. Graph the equation. 3. 2 y x4 3 Graph the equation. 4. y 4x 7 Graphing in Standard Form • One way to graph from standard form is to transform the equation into slope-intercept form. • A second way to graph in standard form is to find the x-intercept and y-intercept. Plot the two points and draw a line through them. x-intercept • The x-intercept is in the form x , 0 . So to find the x-intercept, you will need to substitute 0 into the equation for y and solve for x . y-intercept • The y-intercept is in the form 0 , y . So to find the x-intercept, you will need to substitute 0 into the equation for x and solve for y . • You can use this method to find the x and yintercepts in slope-intercept form as well! Find the x-intercept and y-intercept for the following equations. 5. 3x 2y 12 6. x y 1 5 6 Graph using intercepts. 7. 5x 2y 10 Graph using intercepts. 8. 6x 3y 12 9.Determine the value of k so that the given point will be on the graph of the given equation. 4x 6k 3y; 3,10 Vertical and Horizontal Lines • The equation for a vertical line is in the form: x = # . The slope of a vertical line is undefined . • The equation for a horizontal line is in the form: y = # . The slope of a horizontal line is zero . Graph the Equation 10. y5 Graph the Equation 11. x 3 Absolute Value Graphs • Absolute value graphs are in the shape of a V . Use a table of values to graph the function. 12. y x • On your graphing calculator graph the following functions: • Y1: yx • Y2: y x3 • Y3: y x4 • What happened to the graph? • On your graphing calculator graph the following functions: • Y1: yx • Y2: y x 3 • Y3: y x 2 2 2 2 • What happened to the graph? • On your graphing calculator graph the following functions: • Y1: y x • Y2: y x5 3 • Y3: y x 1 4 • What happened to the graph? 13. Without your calculator, graph the following functions. a. y x 1 2 13. Without your calculator, graph the following functions. y x2 3 b.