lmm open course

1.1 Understanding Linear Equations and Graphs

1.1.1 Intercepts of Lines

• Not every line has two distinct intercepts.

• Example: Horizontal lines do not cross the x-axis and thus have no x-intercept.

• The equation of a horizontal line has the general form

  • Example equation: [ y = c ] where c is a constant

  • The graph is a horizontal line parallel to the x-axis.

1.1.2 Types of Intercepts

• Lines with coinciding intercepts:

  • A line through the origin will have an x-intercept and a y-intercept that are the same.

  • For a line through the origin, we can start by looking for intercepts:

    • Set y = 0 to find the x-intercept.

    • Set x = 0 to find the y-intercept.

1.1.3 Graphing Lines through Intercepts

• To graph a line, we need two points:

  • The intercepts give one point each.

• If only one point is available from intercepts:

  • Choose other values for x or y to find a second point.

1.1.4 Graphs and Examples

• Example 12: Graph of a Line through the Origin

  • Example: y = [3x - 12 ]

    • The x-intercept can be found by setting y to 0 and solving.

    • The y-intercept can be found by setting x to 0 and solving.

1.2 Application of Linear Equations

1.2.1 Setting Up Mathematical Models

• Linear equations provide approximations to real-world situations.

• Example of tuition costs at public colleges over the years:

  • Data in a table illustrates costs for selected years from 2000 to 2009.

1.2.2 Data Representation

• Example 14: Costs plotted on a scatterplot show an approximate linear trend.

  • Although the data isn't perfectly linear, it can be approximated with a linear equation.

• Steps:

  1. Plot the data points on a graph.

  2. Identify two points to derive a linear equation using the slope formula.

  3. Graph both the data points and the derived equation.

1.2.3 Predicting Future Values

• Using the linear equation derived from the model, estimate costs for future years (e.g., 2030).

  • Example 14 discusses using a slope-intercept equation to predict to year 2030.

  • Caution: Predictions far into the future may considerably deviate from the actual values due to various influencing factors (e.g., economic trends, policies).

1.2.4 Use of Technology

• Creating plots and linear models can be efficiently done using graphing calculators.

• Steps to plot using a TI-84:

  1. Store data in lists.

  2. Define the viewing window.

  3. Plot the graph and find intersection points

1.3 Understanding Correlation and Regression

1.3.1 Correlation Coefficient

• Measures the strength of the linear relationship between two variables.

  • Ranges from -1 to 1.

  • Close to 1 indicates a strong positive, and close to -1 indicates a strong negative linear relationship.

  • A value of 0 indicates no linear correlation.

1.3.2 Least Squares Method

• The least squares method finds the best-fitting line through data points by minimizing the sum of the squares of the vertical distances from each point to the line.

• Line Equation Form: ( Y = mx + b ) where m is the slope and b is the y-intercept.

Example of Calculating the Least Squares Line

• Example: Analyzing accidental death rates.

  • Calculate necessary sums from data.

  • Derive the slope (m) and y-intercept (b) to formulate the least squares equation.

  • Predict future values based on the calculated equation while being cautious of extrapolation beyond the data range.

Technology Usage

• Efficient calculation of least squares regression may be supported by calculators and statistics software, which can automate processes to find correlation coefficients.