Algebra 1 Section 6.1

Graphing By recognizing key characteristics of a function, you can quickly determine the shape and location of its graph.

Linear Functions f(x) = 2x + 7 y = 2x + 7 These are called linear functions or linear equations because all the points that satisfy them form a straight line.

Example 1 Graph {(x, y) | y = x + 5} x x + 5 (x, y) -2 -1 1 2 1 2 (-2) + 5 = 3 (-1) + 5 = 4 (0) + 5 = 5 (1) + 5 = 6 (2) + 5 = 7 (-2, 3) (-1, 4) (0, 5) (1, 6) (2, 7)

Example 1 y x

Standard Form The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers. Several different standard-form equations are possible for the same linear function.

Standard Form If a linear equation in standard form contains fractional or decimal coefficients, multiply each term so that A, B, and C become integers. It is also common for A to be nonnegative.

Example 2 5 f(x) = - x + 2 7 5 y = - x + 2 7 5 x + y = 2 7

Example 3 Graph x + 3y = 4. Solve for y: 3y = -x + 4 y = - + 4 3 x

Example 3 y = - + 4 3 x Make a table to find ordered pairs. (0, ), (1, 1), (4, 0) Graph the ordered pairs and connect with a straight line. 4 3

Graphing a Linear Equation Solve the linear equation for y. Make a table of at least three ordered pairs using convenient values for x.

Graphing a Linear Equation Graph the ordered pairs on a Cartesian plane. Connect the points with a straight line.

Definitions The y-intercept is the point where the graph intersects the y-axis. The x-intercept is the point where the graph intersects the x-axis.

Intercepts A line’s intercepts can be found from the equation without graphing. The y-intercept always has zero for its x-coordinate. The x-intercept always has zero for its y-coordinate.

Example 4 2x + y = 6 To find the x-intercept, substitute 0 for y and solve for x. 2x + 0 = 6 2x = 6 x = 3

Example 4 2x + y = 6 To find the y-intercept, substitute 0 for x and solve for y. 2(0) + y = 6 0 + y = 6 y = 6

Example 4 2x + y = 6 Plot each intercept: The x-intercept is (3, 0). The y-intercept is (0, 6). Graph the line. Check by using a third point.

Horizontal and Vertical Lines An equation of the form y = c is a horizontal line. An equation of the form x = c is a vertical line.

Example 5 y = -2 x = 3 y y = -2 regardless of the value chosen for x. x = 3 regardless of the value chosen for y. x

Example 5 y = -2 is a function; it passes the vertical line test. x = 3 is not a function; it fails the vertical line test.

Homework: pp. 235-236