Graphing Linear Equations – Comprehensive Study Notes

Vocabulary and Core Concepts

• Independent Variable (Input)

  • Denoted by $x$.

  • You, the graph-builder, get to choose these values.

• Dependent Variable (Output)

  • Denoted by $y$.

  • Computed from the rule/equation once an $x$ is selected.

• Ordered Pair

  • Written $(x,\,y)$ with $x$ first, comma, then $y$.

  • Represents a single point on the coordinate plane.

• Coordinate Axes

  • Horizontal axis = $x$-axis.

  • Vertical axis = $y$-axis.

• Slope–Intercept Form

  • $y = mx + b$ where

  • $m$ = slope (rate of change, “rise over run”).

  • $b$ = $y$-intercept (point where the graph crosses the $y$-axis; occurs when $x = 0$).

Method 1 – Graphing With a Table of Points

1. Pick several small $x$-values (mix of negatives, zero, and positives).

2. Substitute each $x$ into the equation to find $y$.

3. Record every $(x,y)$ in a table.

4. Plot all points on the grid and connect them with a straight line.

Example 1 – $y = 3x + 2$

Chosen $x$-values: ${-2,-1,0,1,2}$

Plot each point, then draw a straight line through them.

Practice Problem 1 – $y = 2x + 1$ (worked in video)

Connect all five points to complete the graph.

Key takeaway: Small integer $x$-choices make arithmetic/plotting quick and accurate.

Method 2 – Graphing With Slope–Intercept Form

1. Ensure equation is written $y = mx + b$.

2. Plot the $y$-intercept $(0,b)$ first.

3. Use the slope $m$ as a movement rule: $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x}$.

   • Positive rise = up; negative rise = down.

   • Positive run = right; negative run = left.

4. From the $y$-intercept, perform the rise/run repeatedly to generate as many additional points as desired (at least 3 overall for accuracy).

5. Draw a straight line through all plotted points.

Example 2 – $y = 2x + 3$

• $y$-intercept: $b = 3$ ⇒ point $(0,3)$.

• Slope: $m = 2$. Any whole number can be written over $1$, so interpret as $m = \dfrac{2}{1}$.
  • Rise $= 2$ (up 2); Run $= 1$ (right 1).

From $(0,3)$:

1. Up $2$ → $y = 5$, Right $1$ → $x = 1$ ⇒ point $(1,5)$.

2. Repeat: $\uparrow 2$ to $y = 7$, $\rightarrow 1$ to $x = 2$ ⇒ point $(2,7)$.

3. For symmetry, reverse the moves: Down $2$ and Left $1$ from $(0,3)$ ⇒ point $(-1,1)$.

Draw the line through all points.

Practice Problem 2 – $y = 3x + 1$

• $y$-intercept: $(0,1)$.

• Slope: $m = 3 = \dfrac{3}{1}$. Rise $=3$, Run $=1$.

Points generated:

1. From $(0,1)$ → up $3$, right $1$ ⇒ $(1,4)$.

2. Again → $(2,7)$.

3. Reverse direction → down $3$, left $1$ ⇒ $(-1,-2)$.

Plot and connect for final line.

Common Mistakes & How to Avoid Them

1. Sign Slips – Watch negatives when substituting: $2(-2) \neq 4$; it’s $-4$.

2. Axis Mix-Ups – $x$ is always horizontal; $y$ is vertical.

3. Ordered-Pair Order – Write $(x,y)$ not $(y,x)$.

4. Slope Over 1 – Any integer slope can be written as $\frac{n}{1}$ to extract run.

5. Too Few Points – At least three points (including the intercept) ensure an accurate straight line.

Real-World & Foundational Connections

• Linear equations model constant-rate situations: speed over time, cost per item, etc.

• Slope $m$ is identical to “rate of change” you met in earlier arithmetic sequences.

• $y$-intercept corresponds to initial value/start-up cost.

Practical & Ethical Implications

• Accurate graphing builds visual intuition for algebraic relationships—a core skill in science, engineering, economics.

• Ethical research/reporting requires precise graph construction; mis-plotted lines can mislead.

Numerical Summary of Examples

Example 1: $y = 3x + 2$ produced line through $( -2,-4 ), ( -1,-1 ), (0,2), (1,5), (2,8)$.
Example 2: $y = 2x + 3$ used slope $\frac{2}{1}$ and intercept $3$.
Practice 2: $y = 3x + 1$ used slope $\frac{3}{1}$ and intercept $1$.