Chapter 3 Graphs and Functions

Chapter Sections 3.1 – Graphs 3.2 – Functions 3.3 – Linear Functions: Graphs and Applications 3.4 – The Slope-Intercept Form of a Linear Equation 3.5 – The Point-Slope Form of a Linear Equation 3.6 – The Algebra of Functions 3.7 – Graphing Linear Inequalities Chapter 1 Outline

§ 3.1 Graphs

Definitions A graph shows the relationship between two variables in an equation. The Cartesian (rectangular) coordinate system is a grid system used to draw graphs. It is named after its developer, René Descartes (1596-1650).

Definitions y II I x III IV The two intersecting axis form four quadrants, numbered I through IV. The horizontal axis is called the x-axis. The vertical axis is called the y-axis.

Definitions y Origin x (0, 0) The point of intersection of the two axes is called the origin. The coordinates, or the value of the x and the value of the y determines the point. This is also called an ordered pair.

Starting at the origin, move 3 places to the right. Plotting Points Starting at the origin, move 3 places to the right. Plot the point (3, -4). The x-coordinate is 3 and the y-coordinate is –4.

Plotting Points Plot the point (3, -4). Then move 4 places down. Plot the point (3, -4). The x-coordinate is 3 and the y-coordinate is –4.

Plotting Points (3, -4) Plot the point (3, -4). The x-coordinate is 3 and the y-coordinate is –4.

Linear Equations Examples: 4x – 3y = 12 x + 2y = -35 A linear equation in two variables is an equation that can be put in the form ax + by = c where a, b, and c are real numbers. This is called the standard form of an equation. Examples: 4x – 3y = 12 x + 2y = -35

Solutions to Equations The solution to an equation is the ordered pair that can be substituted into the equation without changing the “validity” of the equation. Is (3, 0) a solution to the equation 4x – 3y = 12? 4x – 3y = 12 4(3) – 3(0) = 12 12 – 0 = 12 12 = 12  Yes, it is a solution.

Graphing A graph of an equation is an illustration of the set of points whose ordered pairs are solutions to the equation. A set of points that are in a straight line are collinear. The points (-1, 4), (1, 1) and (4, -3) are collinear.

Graph Nonlinear Equations Equations whose graphs are not straight lines are called nonlinear equations. Example: Graph y = x2 – 4. Use the following values for x: -3, -2, -1, 0, 1, 2, and 3.