Unit 4: Linear Equations, Slope, and Graphing Summary

Rate of Change and Slope

• Rate of change is a rate that shows how one variable changes with respect to another; on a linear graph, this is called the slope.
• Slope is the ratio of vertical change ($\text{rise}$) to horizontal change ($\text{run}$) between any two points on a line:   $m = \frac{\text{rise}}{\text{run}}$
• Slope remains constant between any two points on the same line and is written as a fraction in simplest form.
• The variable used to represent slope is $m$.
• The four types of slope are:
  • Positive
  • Negative
  • Zero
  • Undefined
• The slope formula between two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$ is:   $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Missing coordinates can be found by substituting the known coordinates and the given slope $m$ into the slope formula and solving for the unknown value $x$ or $y$.

Forms of Linear Equations

• Slope-Intercept Form:$y = mx + b$
  • $m$ represents the slope.
  • $b$ represents the y-intercept.
• Standard Form:$Ax + By = C$
• Converting Standard Form to Slope-Intercept Form:
  • Isolate the variable $y$ by subtracting the $Ax$ term from both sides and dividing every term by $B$.

Graphing Linear Equations

• Graphing by Slope-Intercept Form:
  • Write the equation in slope-intercept form ($y = mx + b$).
  • Plot the y-intercept at the point $(0, b)$.
  • Use the slope ratio ($\frac{\text{rise}}{\text{run}}$) from $(0, b)$ to plot additional points.
  • Draw a straight line extending through the points with an arrow on both ends.
• Graphing by Intercepts:
  • x-intercept: The point where the line crosses the x-axis. Find algebraically by setting $y = 0$ and solving for $x$, yielding the coordinate $(x, 0)$.
  • y-intercept: The point where the line crosses the y-axis. Find algebraically by setting $x = 0$ and solving for $y$, yielding the coordinate $(0, y)$.
  • Plot both intercepts on their respective axes and connect them with a straight line.

Vertical and Horizontal Lines

• Vertical Lines:
  • Written in the form $x = a$, where $a$ represents the line's x-intercept.
  • Slope is undefined.
  • Parallel to the y-axis.
• Horizontal Lines:
  • Written in the form $y = a$, where $a$ represents the line's y-intercept.
  • Slope is $0$.
  • Parallel to the x-axis.

Questions & Discussion

• Question: What is the slope of the line $y = 4$?
  • Response: $0$
• Question: What is the slope of the line $x = -2$?
  • Response: undefined
• Question: What is the slope of the line $x = 0$?
  • Response: undefined
• Question: Which axis is $y = -1$ parallel to?
  • Response: x-axis
• Question: Which axis is $x = 4$ parallel to?
  • Response: y-axis
• Question: How does the graph of $y = 3$ differ from $y = 3x$?
  • Response: For $y = 3$, $m = 0$ and $b = 3$. For $y = 3x$, $m = 3$ and $b = 0$. The slopes and intercepts are switched.