6.6 Linear Applications, Functions, and Graphs

From Algebra to Interpretation

Linear applications and graphs: Applying linear equations to real-life contexts, using elementary linear functions to describe relationships, and graphing linear equations in two variables, linear inequalities, parallel and perpendicular lines, and systems of equations.

This area carries 2 to 4 of the 20 QAS questions — tied for the largest single allocation on the test. The same content area appears again on the AAF test, so time invested here pays twice.

Forms of a Linear Equation

In $y = mx + b$: $m$ is the slope (rate of change) and $b$ is the $y$-intercept (value when $x = 0$).

Slope

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}$$

The ordering trap: subtract the coordinates in the same order in numerator and denominator. Using $y_2 - y_1$ over $x_1 - x_2$ produces the wrong sign.

Slope through $(2, -3)$ and $(6, 5)$: $m = \dfrac{5 - (-3)}{6 - 2} = \dfrac{8}{4} = 2$

Interpreting Slope in Context

This is what distinguishes an applications question from a pure algebra question. The answer is not a number — it is a number with units and meaning.

A plumber's charge is modeled by $C = 85 + 60h$, where $h$ is hours.

• Slope 60: the cost increases by $60 per additional hour.
• Intercept 85: the fixed call-out fee, charged even at zero hours.

A tank's volume is $V = 500 - 8t$, where $t$ is minutes.

• Slope −8: the tank loses 8 gallons per minute. The negative sign means decrease.
• Intercept 500: the initial volume.

A question asking "what does 8 represent" wants "the rate at which the tank drains, in gallons per minute" — not "the slope."

Writing an Equation From Context

A car rental costs $45 per day plus a flat $20 insurance charge. Write the cost C for d days.

Fixed value → intercept; per-unit value → slope:

$C = 45d + 20$

A membership costs $310 for 5 months and $490 for 9 months. Write the linear model.

Slope first:

$m = \dfrac{490 - 310}{9 - 5} = \dfrac{180}{4} = 45$ dollars per month

Then use point-slope with $(5, 310)$:

$y - 310 = 45(x - 5) \Rightarrow y = 45x + 85$

The $85$ is a one-time initiation fee. Check with the second point: $45(9) + 85 = 405 + 85 = 490$ ✓

Parallel and Perpendicular Lines

• Parallel: equal slopes, $m_1 = m_2$.
• Perpendicular: slopes are negative reciprocals, $m_1 \cdot m_2 = -1$.

A line perpendicular to $y = \tfrac{3}{4}x + 2$ has slope $-\tfrac{4}{3}$.

Flip the fraction and change the sign. Doing only one of the two is the standard error.

Special case: a horizontal line ($m = 0$) is perpendicular to a vertical line (undefined slope), even though their slopes cannot multiply to $-1$.

Find the line through $(4, 1)$ parallel to $y = -2x + 7$.

Parallel → $m = -2$. Point-slope: $y - 1 = -2(x - 4) \Rightarrow y = -2x + 9$.

Graphing Linear Inequalities

Three steps:

1. Graph the boundary line as though the inequality were an equation.
2. Dashed line for $<$ or $>$ (boundary excluded); solid for $\le$ or $\ge$ (boundary included).
3. Shade the half-plane containing solutions — test $(0,0)$ when it is not on the line.

Graph $y > 2x - 3$.

Boundary $y = 2x - 3$, dashed. Test $(0,0)$: is $0 > 2(0) - 3 = -3$? Yes. Shade the side containing the origin — above the line.

When $y$ is isolated, $>$ shades above and $<$ shades below. But if you multiply or divide an inequality by a negative number, reverse the symbol — then re-check.

Systems of Linear Equations Graphically

The solution is the intersection point.

How many solutions does this system have? $y = 3x - 4$ $6x - 2y = 8$

Rewrite the second: $-2y = -6x + 8 \Rightarrow y = 3x - 4$. Identical lines — infinitely many solutions.

The fast diagnostic is to put both equations in slope-intercept form and compare $m$ and $b$. Same $m$ and same $b$ means one line; same $m$ and different $b$ means no solution; different $m$ means exactly one.

Reading Graphs in Applied Items

Applied graph questions typically ask for one of four things:

1. A value at a point — read the coordinate.
2. A rate — compute the slope over an interval.
3. An intercept meaning — the starting or break-even value.
4. A comparison — which relationship grows faster (steeper slope).

Always read the axis labels and units first. A graph plotting thousands of dollars against months means a slope of 2 is $2,000 per month, and distractors are built from exactly that unit confusion.