Overview
This lecture covers the basics of graphing on the coordinate plane, interpreting graphs, determining intercepts, and using graphing technology to analyze equations and real-world data.
The Coordinate Plane and Plotting Points
- The coordinate plane has a horizontal x-axis and a vertical y-axis that intersect at the origin (0, 0).
- Each point is represented by an ordered pair (x, y), showing position relative to the origin.
- Moving right on the x-axis and up on the y-axis indicates positive values; left and down indicate negatives.
- The plane is divided into four quadrants, numbered I (upper right) through IV (lower right), moving counter-clockwise.
- Points on axes are not considered part of any quadrant.
Graphs and Equations
- Equations with two variables (such as y = 2x + 4) can be graphed as a collection of points that make the equation true.
- An ordered pair is a solution if substituting its values into the equation results in a true statement.
Intercepts
- The x-intercept is where the graph crosses the x-axis (y = 0); solve for x to find it.
- The y-intercept is where the graph crosses the y-axis (x = 0); solve for y to find it.
- To find intercepts, substitute 0 for the other variable and solve the resulting equation.
Graphing and Sketching Functions
- Plot intercepts and additional points to sketch the curve (e.g., quadratic functions form a U shape).
- Use a table of values for selected x-values to find corresponding y-values.
Interpreting Graphs
- Analyze real-world graphs by identifying maximum and minimum values, trends, and differences over time.
- For example, find the peak or compare values between years to determine changes.
Using Graphing Technology
- Enter equations into calculators or software (preferably solved for y).
- Adjust the window (x-min, x-max, y-min, y-max, scales) to view relevant graph portions.
- Use calculator functions to find zeros (x-intercepts) and extrema (maximum/minimum points).
Solving and Analyzing with Technology
- The zeros of a function (where y = 0) correspond to solutions of the equation.
- Set left and right bounds to isolate intercepts or extrema then let the calculator approximate the value.
Key Terms & Definitions
- Coordinate Plane — a plane with perpendicular x- and y-axes used to graph points.
- Origin — the point (0, 0) where axes intersect.
- Quadrants — the four sections made by the axes on the plane.
- Ordered Pair — (x, y), a pair of numbers representing point location.
- Intercepts — where a graph crosses the axes; x-intercept (y=0), y-intercept (x=0).
- Zero — an x-value where the function equals zero (x-intercept).
- Extremum — maximum or minimum point on a graph.
Action Items / Next Steps
- Practice identifying and plotting intercepts for given equations.
- Use a graphing calculator or software to sketch and analyze functions.
- Complete exercises on finding solutions to equations using graphing technology.