Graphing is a fundamental concept in mathematics that allows us to visually represent and analyze data, functions, and relationships between variables. In this study guide, we will explore the basics of graphing, including plotting points, graphing linear equations, and interpreting graphs.

When graphing, we often start by plotting points on a coordinate plane. The coordinate plane consists of two perpendicular number lines, called the x-axis and y-axis. The point where the x-axis and y-axis intersect is called the origin, and is typically denoted as (0, 0).

To plot a point, we use an ordered pair (x, y), where x represents the horizontal position on the x-axis and y represents the vertical position on the y-axis. For example, to plot the point (2, 3), we move 2 units to the right along the x-axis and 3 units up along the y-axis, and then mark the point where these two movements intersect.

A linear equation is an equation that represents a straight line on a graph. The standard form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis).

To graph a linear equation, we can use the y-intercept to plot the first point on the graph, and then use the slope to find additional points. The slope m represents the change in y divided by the change in x, so if m = 2/3, for example, we would move 2 units up and 3 units to the right to find the next point on the line.

Once a graph is created, we can interpret the information it presents. We can analyze the slope of a line to understand the rate of change, identify key points such as intercepts and turning points, and compare multiple graphs to analyze relationships between different variables.

- Plot the points (1, 2), (3, 4), and (5, 6) on a coordinate plane.
- Graph the equation y = 2x - 3.
- Interpret the graph of y = 3x + 2 in terms of its slope and y-intercept.

By mastering the concepts of plotting points, graphing linear equations, and interpreting graphs, you will develop a strong foundation in graphing that will be essential for further studies in algebra, geometry, and beyond.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.