Section 1.1: Equations of Lines

Understanding Linear Equations

Linear equations describe straight lines in the Cartesian plane and are fundamental in calculus and algebra. The general form is:

• Equation of a line:

• Slope (m): Measures the steepness of the line; calculated as the change in y over the change in x between two points.

• Y-intercept (b): The value of y where the line crosses the y-axis (when ).

Key Skills:

• Find the equation of a line given a point and a slope, or two points.

• Equations of horizontal lines: (slope ).

• Equations of vertical lines: (undefined slope).

• Find equations of lines parallel (same slope) or perpendicular (negative reciprocal slope) to a given line.

• Use the equation of a line to make predictions (e.g., extrapolation/interpolation).

Example: Find the equation of the line passing through with slope .

• Use point-slope form:

• Simplify:

Section 1.2: Best Fit Lines and Correlation

Regression and Correlation

Best fit lines are used in statistics to model the relationship between two variables. The line minimizes the sum of squared vertical distances from the data points to the line (least squares method).

• Purpose of a best fit line: To model and predict the relationship between variables.

• Minimized Quantity: The sum of squared residuals (vertical distances).

• Mean (, ): Average of x and y values.

• Variance:

• Covariance:

• Slope of best fit line:

• Intercept:

• Correlation (r):

• Interpretation of r: close to 1 implies a strong linear fit; close to 0 implies a weak fit.

• Positive correlation: As x increases, y increases.

• Negative correlation: As x increases, y decreases.

Example: Given data points, compute the best fit line and correlation coefficient to assess the linear relationship.

Section 1.3: Functions and Their Properties

Definition and Notation

A function is a rule that assigns to each input exactly one output. Functions are often written as instead of .

• Function notation: means the output when input is .

• Evaluating functions: Substitute the input into the function rule.

• Domain: All possible input values (x-values) for which the function is defined.

• Range: All possible output values (y-values) the function can produce.

• Vertical Line Test: A graph represents a function if any vertical line crosses it at most once.

Example: For , the domain is all real numbers, and the range is .

Section 1.4: Quadratic Functions

Key Features and Graphing

Quadratic functions have the form and their graphs are parabolas.

• x-intercepts: Solve for (roots).

• y-intercept:

• Vertex: ,

• Plot intercepts and vertex to sketch the parabola.

• Quadratics can model maximum/minimum problems (e.g., projectile motion, area optimization).

Example: For , vertex at , ; y-intercept at $3x = 1x = 3$.

Section 1.5: Polynomials and Rational Functions

Classification and Properties

• Polynomial: An expression of the form with non-negative integer exponents.

• Degree: The highest exponent of ; determines the end behavior and number of roots.

• Even degree: Both ends of the graph go in the same direction (up or down).

• Odd degree: Ends go in opposite directions.

• Given a graph, estimate the degree by counting the number of turning points (max number is degree minus one).

• Rational function: A ratio of two polynomials, .

• Undefined where (vertical asymptotes).

Example: is undefined at .

Section 2.1: Exponential Functions

Growth and Decay Models

• Exponential function: where is the starting value and is the base (growth if , decay if ).

• Exponential growth:

• Exponential decay:

• Doubling time form: , where is the doubling period.

• Half-life form: , where is the half-life.

• Given a word problem, identify , , and to write the equation and solve for unknowns.

Example: A population doubles every 3 days: .

Section 2.2: Logarithmic Functions

Definition and Properties

• Logarithm: means .

• Evaluate basic logarithms: because .

• Inverse property: and .

• Use logarithms to solve for exponents in doubling/half-life problems.

• Log properties:

• Change of base: for any base .

• Natural logarithm: , where .

• Inverse property: .

Example: Solve by taking logarithms: .

Section 2.3: Applications of Logarithms and Exponentials

Solving Exponential Equations

• Apply logarithm rules to solve equations involving exponentials.

• Applications include carbon dating, bacterial growth, and Newton's law of cooling.

• General approach: Take logarithms of both sides to solve for the exponent.

Example: If , solve for given , , and :

• Take natural log:

Additional info: Newton's law of cooling and other applications will provide the specific formula as needed.