Equations of Lines and Linear Modeling in Precalculus

Equations of Lines and Linear Modeling

Objectives

• Determine equations of lines.

• Given the equations of two lines, determine whether their graphs are parallel or perpendicular.

• Model a set of data with a linear function.

• Fit a regression line to a set of data and use the line model to make predictions.

Slope-Intercept Equation

The slope-intercept equation is a fundamental way to express the equation of a straight line in the Cartesian plane. It is given by:

• General form: $y = mx + b$

• Where:

  • m is the slope of the line (rate of change of y with respect to x)

  • b is the y-intercept (the value of y when x = 0)

• If the slope and y-intercept are known, substitute their values into the equation to obtain the specific line.

Example 1

• Given: Slope $m = -\frac{7}{9}$, y-intercept $(0, 16)$

• Equation: $y = -\frac{7}{9}x + 16$

• Alternate notation: $f(x) = -\frac{7}{9}x + 16$

Example 2

• Given: Slope $m = -\frac{2}{3}$, point $(-3, 6)$

• Substitute into $y = mx + b$:

• $6 = -\frac{2}{3}(-3) + b$

• $6 = 2 + b \implies b = 4$

• Equation: $y = -\frac{2}{3}x + 4$ or $f(x) = -\frac{2}{3}x + 4$

Point-Slope Equation

The point-slope equation is useful when the slope and a point on the line are known. The equation is:

• $y - y_1 = m(x - x_1)$

• Where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Example 3

• Find the equation of the line through $(2, 3)$ and $(1, -4)$.

• Calculate slope: $m = \frac{-4 - 3}{1 - 2} = \frac{-7}{-1} = 7$

• Using point-slope form with $(2, 3)$: $y - 3 = 7(x - 2)$

• Simplify: $y = 7x - 11$ or $f(x) = 7x - 11$

Parallel Lines

Parallel lines are lines in the same plane that never intersect. Their slopes are equal.

• Vertical lines (e.g., $x = a$) are parallel to each other.

• Nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

Examples (from diagrams)

• $y = 2x + 4$ and $y = 2x - 3$ are parallel (same slope, different intercepts).

• $y = 3x - 2$ and $y = -2x + 5$ are not parallel (different slopes).

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). Their slopes are negative reciprocals.

• Lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1 m_2 = -1$.

• Vertical and horizontal lines are also perpendicular (e.g., $x = a$ and $y = b$).

Examples (from diagrams)

• $y = -\frac{1}{2}x + 3$ and $y = 2x - 4$ are perpendicular ($-\frac{1}{2} \times 2 = -1$).

• $y = 3x + 1$ and $y = 4x - 2$ are not perpendicular ($3 \times 4 = 12$).

Mathematical Models

A mathematical model uses mathematical language and equations to represent real-world phenomena. Models allow predictions and analysis of real situations.

• If predictions are inaccurate, the model must be revised or discarded.

• Mathematical modeling is an ongoing process, often involving refinement.

Curve Fitting

Curve fitting is the process of finding a function that best fits a set of data points. Linear curve fitting is a common approach in modeling relationships between variables.

• Scatterplots are used to visualize data and assess if a linear model is appropriate.

Example 6

• Cost of a 30-second Super Bowl commercial increased from 2010 to 2015.

• Model data with a linear function using two points: $(1, 3.1)$ and $(4, 4.0)$.

• Find slope: $m = \frac{4.0 - 3.1}{4 - 1} = \frac{0.9}{3} = 0.3$

• Use point-slope form with $(1, 3.1)$: $y - 3.1 = 0.3(x - 1)$

• Estimate cost in 2018 ($x = 8$): $y = 0.3(8) + 2.8 = 5.2$ million

Linear Regression

Linear regression is a statistical method for modeling the relationship between two variables by fitting a linear equation to observed data.

• Regression line: $y = mx + b$

• Graphing calculators can compute the regression line and display scatterplots.

• Example regression equation: $y = 0.3142857143x + 2.847619048$

• Prediction for $x = 8$: $y = 0.3142857143 \times 8 + 2.847619048 \approx 5.3$ million

Correlation Coefficient

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables.

• $-1 \leq r \leq 1$

• $|r|$ close to 1 indicates strong correlation; close to 0 indicates weak correlation.

• Positive $r$ means positive slope; negative $r$ means negative slope.

Interpretation Table

Value of r	Interpretation
r = 1	Perfect positive correlation (all points on regression line)
r = 0.91	High positive correlation
r = 0.42	Low positive correlation
r = -1	Perfect negative correlation
r = -0.91	High negative correlation
r = -0.42	Low negative correlation

Summary: This section covers the equations of lines (slope-intercept and point-slope forms), criteria for parallel and perpendicular lines, mathematical modeling, curve fitting, linear regression, and the interpretation of the correlation coefficient. These concepts are foundational for understanding linear relationships and modeling in precalculus.