Linear & Proportional Relationships – Complete Study Notes

Foundational Vocabulary and Core Ideas

Key terms introduced

1. Constant of Proportionality
   • Definition – the unchanging value of the ratio between two quantities that vary together.
   • Algebraic form – $k = \frac{y}{x}$ whenever $y$ varies directly with $x$.
   • Significance – acts as the “multiplier” telling us how much one variable grows when the other grows by one unit.

2. Proportional Relationship
   • A set of ordered pairs $(x, y)$ whose ratios $\frac{y}{x}$ are all equal (i.e.
   every pair reduces to the same number $k$).
   • Graphical hallmark – a straight line through the origin $(0,0)$.
   • Equation form – $y = kx$ (or $y = px$; the letter is arbitrary but represents the same constant).

3. Linear Relationship
   • Any two-variable relationship whose graph is a straight line.
   • General equation – $y = mx + b$ where $m$ is slope and $b$ is $y$-intercept.
   • Note: All proportional relationships are linear, but not all linear relationships are proportional (because a proportional line must have $b = 0$).

4. Linear vs. Non-Linear (quick visual test)
   • Ask two questions: (a) Does the graph bend? (b) Does it curve?
   • “No” to both ⇒ linear.
   • “Yes” to at least one ⇒ nonlinear.

5. Exponent Rule for Linearity
   • In standard form, each variable must appear only to the first power.
   • Hidden exponent of $1$ is never written (e.g.
   $y = 2x - 4$ has $x^1$).
   • Exponents other than $1$ (positive, negative, fractional, or radical form) immediately make the relationship nonlinear.

Recognizing Linear vs. Non-Linear Expressions & Graphs

Visual examples

• A wavy or curved trace ⇒ nonlinear.
• A straight slanted segment (even if it doesn’t pass through the origin) ⇒ linear.

Algebraic examples

1. $y = 7x$
   • $x$ appears to the first power ⇒ linear.

2. $y = x^5$
   • $x$ has exponent $5$ ⇒ nonlinear (actually a polynomial of degree 5).

3. $y - \frac{x}{9} = 0$
   • Re-arrange to $y = \frac{x}{9}$; exponent of $x$ is $1$ ⇒ linear.

4. $y = 2x^3$
   • Exponent $3$ ⇒ nonlinear.

5. $y = 2^x$
   • Variable in the exponent (exponential growth) ⇒ nonlinear.

6. Radicals such as $y = \sqrt{x}$ (equivalent to $x^{\frac12}$) ⇒ nonlinear.

Real-World Linear Scenarios

1. Car-wash earnings
   • Base tip: $\$15$.
   • Additional $\$25$ earned per car washed.
   • Equation: $y = 25x + 15$ where $x$ = number of cars, $y$ = total money.
   • Straight-line growth of $25$ dollars each additional car ⇒ linear.

2. Driving distance
   • Speed: $80\ \text{mi/h}$.
   • After $2$ h, distance $=80(2)=160$ mi.
   • Equation: $d = 80t$ (through origin) ⇒ proportional and linear; constant of proportionality $k = 80$.

3. Grocery-store soda promotion
   • Prices: 1 for $\$6$, 2 for $\$12$, 3 for $\$15$.
   • Expected linear pattern would be $18$ dollars for 3 cans; deviation to $15$ breaks constancy.
   • Graph is curved ⇒ nonlinear (possible bulk-discount curve).

Revisiting Proportionality Conditions

1. Required equation form
   • $y = kx$ (no extra constant term).
   • Example: $y = 5x$ (here $k=5$).

2. Origin test
   • Substitute $x = 0$.
   • If $y = 0$ follows, the relation might be proportional; otherwise, definitely not.

3. Sample comparison
   • $y = 5x + 2$
   – Linear but plugging $x=0$ gives $y = 2$ ⇒ graph crosses $y$-axis at $2$, so not proportional.

Table Method for Detecting Proportionality

Step-by-step algorithm:

1. List all ordered pairs $(x<em>i, y</em>i)$.

2. Compute $\frac{y<em>i}{x</em>i}$ for every pair.

3. Compare the quotients:
   • If every quotient equals the same constant $k$ ⇒ proportional relationship (and $k$ is the constant of proportionality).
   • If any quotient differs ⇒ non-proportional.

Caution – do not stop after two matching ratios; the entire set must match to declare proportionality.

Worked Table Examples

Example A (Proportional)

• Pairs: $(2,8),(3,12),(8,32),(10,40)$.
• Ratios: $\frac82 = 4$, $\frac{12}{3}=4$, $\frac{32}{8}=4$, $\frac{40}{10}=4$
• All equal ⇒ constant $k = 4$ ⇒ proportional and linear (passes through origin).

Example B (Not proportional)

• Pairs: $(1,3),(2,5),(3,7),(4,9)$.
• Ratios: $3/1 = 3$, $5/2 = 2.5$ … already inconsistent ⇒ non-proportional.
• The graph will still be a line because the equation is $y = 2x + 1$ (slope $2$, intercept $1$), but not through origin.

Three Comprehensive Scenarios

1. Apple trees vs. apples harvested
   • Data: (2,26), (3,39), (6,78), (10,130).
   • Divisions: $26/2 = 13$, $39/3 = 13$, $78/6 = 13$, $130/10 = 13$.
   • Constant $k=13$ ⇒ proportional; equation $y = 13x$.

2. Movie-ticket pricing
   • One ticket $=\$2.50$, two $=\$5.00$, three $=\$7.50$.
   • Ratios: $2.50/1=2.5$, $5/2=2.5$, $7.50/3=2.5$.
   • Constant $k=2.5$ ⇒ proportional; equation $y = 2.5x$.
   • Practical note – linear, fair pricing per ticket.

3. Bowling-alley costs
   • Base shoe rental $=\$10$ (fixed).
   • Each game $=\$5$.
   • Equation: $y = 5x + 10$ with $x$ = games.
   • Because of the “+ 10” term, $y\neq 0$ when $x=0$ ⇒ linear but not proportional.
   • Ethical/business implication – a fixed fee plus unit cost model is common (membership + usage).

Conceptual Connections & Broader Significance

• Linear functions model countless everyday phenomena (hourly wages, constant-speed travel, pay-per-use services).
• Proportional relationships are a special subset capturing pure “per-unit” situations with no start-up cost—important for spotting fairness or best-buy comparisons.
• Nonlinear patterns (exponential, polynomial, radical) often indicate accelerating change, diminishing returns, or volume/area effects—critical in finance, biology, and physics.

Quick Diagnostic Checklist

1. Inspect the equation:
   • Any exponent other than $1$ on a variable? ⇒ nonlinear.
   • Extra constant term $b \neq 0$? ⇒ not proportional (though possibly still linear).

2. Inspect the graph:
   • Straight and passes the origin? ⇒ proportional.
   • Straight but intercept $\neq 0$? ⇒ linear, non-proportional.
   • Curved? ⇒ nonlinear.

3. Inspect a data table:
   • Compute every $y/x$. All equal? ⇒ proportional; else non-proportional.
   • If differences are constant ((\Delta y/\Delta x = m)), still linear but not proportional.

Summary

• Linearity demands a constant additive rate of change (slope).
• Proportionality demands both constant multiplicative rate and zero intercept.
• Graphical, algebraic, and tabular tests give equivalent verdicts.
• Real-world costs can mix fixed fees and per-unit charges; recognise these as linear but not proportional.