Graphing Linear Equations Algebra Review Graphing Linear Equations 4/6/99 Graphing Linear Equations by Joyce

Variables of an Equation (Review) x and y are most commonly used as variables in an equation Variables are placeholders for values Variable y is dependent on the variable x The independent variable, x, is plotted along the horizontal axis (x axis) The dependent variable, y, is plotted along the vertical axis (y axis) 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Graphing Equations The graph of an equation consists of all the points which satisfy or solve the equation. Solve the equation to find the points,then plot the points. These points are called coordinates. Each point (coordinate) consists of one value for y and one value for x, and is written as the ordered pair (x, y). The values of (x, y) solve the equation y = x + c where c represents a constant value The coordinates are ordered pairs of real numbers. 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce A linear equation is a first degree equation (the highest power exponent is 1) Examples: y = x + 2, 3y = 4x + 2, y = mx + b, and Ax + By + C = 0 where A an B are not equal to 0 The graph of a linear equation is a non vertical straight line Example: y = 2x + 3 is the graph of a straight line 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Linear Graphs Linear graphs are drawn by solving the equations and plotting the points All the points on the graph solve the equation Only two points are needed to graph a straight line Find a third point as a check for accuracy Draw the graph through the points 4/6/99 Graphing Linear Equations by Joyce

Solve for Ordered Pairs Consider the linear equation y = x + 2 The y value is dependent on the value of x Find ordered pairs by assigning any value to x Let x = 0 and solve for y to find (0, y) y = 0 + 2 or y = 2 The ordered pair is (0, 2) Let x = 1 and solve again for y to find (1, y) y = 1 + 2 or y = 3 The ordered pair is (1, 3) Let x = -2 and solve again for y to find (-2, y) y = -2 + 2 or y = 0 The ordered pair is (-2, 0) 4/6/99 Graphing Linear Equations by Joyce

Table and Graph for Ordered Pairs Solutions for y = x + 2 put in a table form x y 0 2 1 3 -2 0 -1 1 Graph of y = x + 2 4/6/99 Graphing Linear Equations by Joyce

Solve y = 2x + 3 for Ordered Pairs Set up a chart and substitute values for x, solving for y Let x = 0 y = 2(0) + 3 y = 3 Let x = 1 y = 2(1) + 3 y = 5 Let x = -1 y = 2(-1) + 3 y = 1 y is dependent on x Chart of Ordered Pairs x y pair 0 3 (0, 3) 1 5 (1, 5) -1 1 (-1, 1) 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Graph of y = 2x + 3 4/6/99 Graphing Linear Equations by Joyce

Slope-Intercept Formula The slope-intercept formula for a straight line is y = mx + b m and b are constant real numbers m is the slope of the line if m is positive, the line slopes up if m is negative, the line slopes down 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Intercepts of y = mx + b The y intercept of a line y = mx + b is the point where it crosses the y axis (the point where x = 0). Let x = 0 and solve for y the x coordinate of this point is, by definition, 0 since the y axis represents the zero value for x The x intercept of a line y = mx + b is the point where it crosses the x axis (the point where y = 0). Let y = 0 and solve for x the y coordinate of this point is, by definition, 0 since the x axis represents the zero value for y 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Slope of y = mx + b If the slope of a line is 3 and the y intercept is -2, then the equation of the line becomes y = 3x - 2 The ordered pair representing the y intercept is (0, -2) The y intercept is found by solving for y when x = 0. So when x = 0, y = 3(0) -2 or y = -2 The slope, m, equals 3 means that there is a change in x of 1 unit for each change in y of 3 units. x goes from 0 to 1 (add 1 to the x value of the y intercept) y goes from -2 to 1 (add 3 to the y value of the y intercept) Another ordered pair on the graph is (1, 1). 4/6/99 Graphing Linear Equations by Joyce

Point-Slope Formula (y1- y2) = m (x1- x2) Given any two points on a line, (x1, y1) and (x2, y2) m is the difference between the y values divided by the difference between the x values or m is the ratio of the change in y coordinates to the change in x coordinates The slope of a vertical line is undefined The denominator, xi - xj, would have the value of zero or 4/6/99 Graphing Linear Equations by Joyce

Tables, Graphs, Equations For a straight line we can make a table from a graph or an equation a graph from an equation or a table an equation from a graph or a table Use the difference in y values divided by the difference in x values (point-slope formula) Use the slope-intercept formula 4/6/99 Graphing Linear Equations by Joyce

Graph from an Equation Graph a line from an equation Solve for the y intercept Solve for the x intercept Draw a line through both Find a third point to check the accuracy Write the equation of a line find two points on the line and substitute the x and y values into the point-slope formula or find two points on the line and substitute the x and y values into the slope-intercept formula 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Graphing Guidelines For the following exercises, you will need paper to sketch the axes as a vertical and a horizontal line. Squared paper is ideal. Lined paper is OK, too. Count the printed lines as increments of one unit up and down, and sketch in the same increments side to side. Even plain paper works: a ruler is useful and you could use 1 cm increments to mark off each axis. These are only sketches, DO NOT take measurements from the paper as a draftsman might. You will need at least 7 increments in each direction of both axes. 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Linear Exercises 1. Plot the points (0, 0) “the origin”, and (4, 3) i.e., x = 4 and y = 3. Draw a line segment between them. Draw another line segment from (4, 3) vertically to (4, 0). What are the lengths of each side of the resulting triangle? Hint: The figure is a right triangle and you can apply the Pythagorean Theorem. 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce More Exercises 2. On the same set of axes as Exercise 1, plot the points (2, 1), (-2, -2), and (2, -2). Connect these three points with line segments. What is different? 3. On a new set of axes, plot the points (0, 0) and (3, 4). Connect these points. Is this the same line segment as in Exercise 1? Plot the point (-3, 4). The slope of a line from (0, 0) to (3, 4) is , that is . What would be the slope of a line from (0, 0) to (-3, 4)? 4/6/99 Graphing Linear Equations by Joyce

Graphing Linear Equations by Joyce Even More Exercises 4. On a new set of axes, plot the points (6, -6), (-6, -1), and (6, -1). 5. Find the distances between the points. 6. Find the slope of the line between the first two points. 4/6/99 Graphing Linear Equations by Joyce