Exam 4

Graphs and the Coordinate Plane

Understanding the Cartesian Coordinate System

The Cartesian coordinate system is fundamental in precalculus for graphing equations, plotting points, and visualizing solution sets. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0).

• Axes: The x-axis and y-axis divide the plane into four quadrants.

• Origin: The point (0,0) where the axes intersect.

• Quadrants: Numbered I to IV, starting from the upper right and moving counterclockwise.

• Plotting Points: Each point is represented as an ordered pair (x, y).

• Example: The point (3, -2) is 3 units to the right of the origin and 2 units down.

Graphing Equations and Inequalities

Graphing Linear Equations

Linear equations in two variables, such as $y = mx + b$, produce straight lines on the coordinate plane. The slope (m) and y-intercept (b) determine the line's steepness and position.

• Slope-Intercept Form: $y = mx + b$

• Slope: $m = \frac{\Delta y}{\Delta x}$

• Y-intercept: The point where the line crosses the y-axis ($x=0$)

• Example: $y = 2x + 1$ has slope 2 and y-intercept 1.

Graphing Points and Solution Sets

Individual points, lines, and shaded regions on the coordinate plane can represent solutions to equations or inequalities.

• Single Point: Represents a unique solution, e.g., (4,5).

• Line: Represents all solutions to a linear equation.

• Shaded Region: Represents all solutions to an inequality.

• Example: The region above the line $y = x$ represents solutions to $y > x$.

Graphing Linear Inequalities

Solution Sets for Inequalities

Linear inequalities such as $y > x$ or $x + y \leq 3$ are graphed by shading the region of the plane that satisfies the inequality. The boundary line is drawn solid for $\leq$ or $\geq$, and dashed for $<$ or $>$.

• Boundary Line: Graph the related equation as a line.

• Test Point: Use a point not on the line (often the origin) to determine which side to shade.

• Shading: The solution set is the region where the inequality holds true.

• Example: For $y \leq 2x - 1$, shade below the line $y = 2x - 1$.

Systems of Equations and Inequalities

Graphical Solutions to Systems

Systems of equations or inequalities involve finding points or regions that satisfy all given conditions simultaneously.

• Intersection Point: For two lines, the intersection is the solution to both equations.

• Overlapping Region: For inequalities, the solution set is the region where all shaded areas overlap.

• Example: The solution to $y > x$ and $y < 2x + 1$ is the region between the two lines.

Tables: Types of Graphical Solution Sets

Graph Type	Represents	Example
Point	Single solution	(2, 3)
Line	All solutions to a linear equation	$y = -x + 4$
Shaded Region	All solutions to a linear inequality	$y \geq x - 2$
Overlapping Shaded Regions	Solutions to a system of inequalities	$y > x$, $y < 2x + 1$

Additional info:

• Some images show only axes and points, which are used for plotting solutions or demonstrating the location of solutions to equations.

• Shaded regions in the images indicate solution sets to inequalities or systems of inequalities.

• Dashed or solid lines in the graphs correspond to strict ($<$, $>$) or inclusive ($\leq$, $\geq$) inequalities, respectively.