Notes on Functions, Intercepts, and Graphing

Introduction to Functions

• Understand the basic types of functions:
  • Increasing Functions: As $x$ increases, $y$ also increases.
  • Decreasing Functions: As $x$ increases, $y$ decreases (negative slope).
  • Constant Functions: As $x$ increases, $y$ remains the same (horizontal line).

Intercepts

• Y-Intercept: The value of $y$ when $x = 0$.
  • Example: If $x = 0$ in function $y = mx + b$, then $y = b$.
• X-Intercept: The value of $x$ when $y = 0$.
  • To find the x-intercept, set $y = 0$ and solve for $x$.
• Finding Intercepts:
  • Use slope-intercept form $y = mx + b$ to find intercepts easily.
  • Common mistake: Confusing the x-intercept and y-intercept as a single ordered pair; they are distinct.

Example of Finding Intercepts

• Given the function $3x + 0y = 12$:
  1. For $y$-intercept: Set $x = 0$;
  • $3(0) + 0y = 12$
  • This leads to no $y$ value, identifying point as (0,4).
  1. For $x$-intercept: Set $y = 0$;
  • $3x + 0 = 12$
  • Solve: $x = 4$ (points: (4,0)).

Characteristics of Lines

• Lines are defined by two points for accuracy; a line is straight, regardless of inclination.
  • Use straight edges (like the edge of a notebook) for accurate lines, not curved.

Working with Functions

• Consider function example: $y = -2x + 4$
• Create a Table of Values: Choose values for $x$ and calculate corresponding $y$ values:
  1. If $x = 0$: $y = -2(0) + 4 = 4$
  2. If $y = 0$: Solve:
  • $0 = -2x + 4$;
  • Rearranging, $2x = 4$ leads to $x = 2$.
• Plot points found: (0, 4) and (2, 0) on a graph.
  • Ensure to draw a line with arrows indicating the direction of the line continues infinitely.

Additional Points and Verification

• It’s beneficial to use at least three points to verify accuracy when graphing.
• Understand that negative coefficients will lead to decreasing trends in line graphs.
• An example function $y = \frac{1}{2}x - 2$ helps illustrate more complex linear functions.
  • Find intercepts:
    • When $x = 0$: $y = -2$
    • When $y = 0$: Rearranging leads to $x = 4$.
• Plot both: (0, -2) and (4, 0).

Exam Preparation

• Concepts reviewed will be important for upcoming exam; practice plotting and solving functions to prepare.
• Clarity and precision matter in solving linear functions and intercept identification.