Linear Functions: Identifying Slopes, Intercepts, and Graphing Procedures

Analysis of the Linear Function f(x) = 3x - 2

• The provided transcript outlines a two-part mathematical problem involving the analysis and graphing of a specific linear function.
• The function presented is defined as:     $f(x) = 3x - 2$
• This equation is written in the slope-intercept form, which is standard for representing linear relationships. The general equation is:     $f(x) = mx + b$
• In this general equation:
  • $m$ is the variable representing the slope of the line.
  • $b$ is the variable representing the y-intercept of the line.

Part A: Identification of Slope and Y-Intercept

• The objective of Part A is to explicitly identify the numerical values for the slope and the y-intercept based on the equation $f(x) = 3x - 2$.

• Identification of the Slope ($m$):

  • The slope is the coefficient of the independent variable $x$.
  • In the function $f(x) = 3x - 2$, the coefficient is $3$.
  • Therefore, the value to be entered for "Slope =" is $3$.
  • Mathematical Meaning: A slope of $3$ indicate that the "rise over run" is $\frac{3}{1}$. For every $1$ unit increase in $x$, the value of $f(x)$ increases by $3$ units.

• Identification of the Y-Intercept ($b$):

  • The y-intercept is the constant term added to or subtracted from the $mx$ term.
  • In the function $f(x) = 3x - 2$, the constant term is $-2$.
  • Therefore, the value to be entered for "y-intercept =" is $-2$.
  • Mathematical Meaning: The y-intercept represents the point where the line crosses the vertical y-axis. This occurs when $x = 0$. Consequently, the coordinates for this point are $(0, -2)$.

Part B: Graphing the Line in the Coordinate Plane

• The objective of Part B is to translate the algebraic function onto a provided coordinate plane.

• Coordinate Plane Characteristics:

  • The x-axis (horizontal) is shown with a range of $[-10, 10]$, with highlighted markers at intervals of $2$ ($-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10$).
  • The y-axis (vertical) is shown with a range of $[-10, 10]$, also with highlighted markers at intervals of $2$.

• Step-by-Step Graphing Procedure:

  1. Plot the Y-Intercept: Begin by identifying the point $(0, -2)$ on the y-axis. Place a point at this location.
  2. Apply the Slope (Rise over Run): Using the slope of $3$ (or $\frac{3}{1}$), move from the y-intercept at $(0, -2)$:
     • Move up $3$ units in the positive y-direction.
     • Move right $1$ unit in the positive x-direction.
     • The new point will be at $(1, 1)$. Plot this point.
  3. Generate Additional Points: Repeat the slope application to ensure accuracy:
     • From $(1, 1)$, move up $3$ units and right $1$ unit to reach $(2, 4)$. Plot this point.
     • From $(2, 4)$, move up $3$ units and right $1$ unit to reach $(3, 7)$. Plot this point.
     • To find points in the negative direction, move down $3$ units and left $1$ unit from the y-intercept $(0, -2)$ to reach $(-1, -5)$. Plot this point.
  4. Draw the Line: Connect the plotted points with a solid, straight line that extends across the entire coordinate plane, typically adding arrows at the ends to show the line continues infinitely.

Summary of Key Values for Question 23

• Original Expression: $f(x) = 3x - 2$
• Slope: $3$
• Y-Intercept: $-2$
• X-Intercept calculation (Optional detail):
  • Set $f(x) = 0$
  • $0 = 3x - 2$
  • $2 = 3x$
  • $x = \frac{2}{3}$
  • The line will cross the horizontal x-axis at the point $(\frac{2}{3}, 0)$.