Linear Functions Pattern - Data to Equation

This pattern asks you to find the equation of a linear function from given information — a table of values, two points, or a slope with a point. Every question boils down to identifying the slope $m$ and y-intercept $b$ to write $f(x) = mx + b$.

 

Given Slope and Y-Intercept Directly

The simplest version gives you the slope and a point where $x = 0$ (which is the y-intercept). Just plug straight into $f(x) = mx + b$.

For a linear function $g$, the graph of $y = g(x)$ in the xy-plane has a slope of $-2$ and passes through the point $(0, 7)$. Which equation defines $g$?

A) $g(x) = 2x + 7$
B) $g(x) = -2x + 7$
C) $g(x) = -2x - 7$
D) $g(x) = -2x + \dfrac{1}{7}$

Since the point is $(0, 7)$, the y-intercept is $b = 7$. The slope is $m = -2$. So $g(x) = -2x + 7$. The answer is B.

When the line passes through the origin $(0, 0)$, the y-intercept is $0$ and the equation simplifies to $f(x) = mx$.

For the function $q$, the graph of $y = q(x)$ in the xy-plane is a line that passes through the origin. If the slope of the line is $-\dfrac{4}{3}$, which equation defines $q$?

A) $q(x) = -\dfrac{3}{4}x$
B) $q(x) = \dfrac{4}{3}x$
C) $q(x) = -\dfrac{4}{3}x$
D) $q(x) = -\dfrac{4}{3}x + 1$

Origin means $b = 0$, so $q(x) = -\dfrac{4}{3}x$. The answer is C. Watch for the reciprocal trap (option A) and the sign flip (option B).

 

Given Two Function Values (One at x = 0)

Many Easy questions give two values like $f(0) = 15$ and $f(1) = 20$. Since one input is $0$, you get the y-intercept for free, then compute the slope.

A linear function $g$ is defined such that $g(0) = 15$ and $g(1) = 20$. Which equation defines $g$?

A) $g(x) = 5x$
B) $g(x) = 15x$
C) $g(x) = 5x + 15$
D) $g(x) = 15$

From $g(0) = 15$, the y-intercept is $b = 15$. The slope is $m = \dfrac{20 - 15}{1 - 0} = 5$. So $g(x) = 5x + 15$. The answer is C.

A special case: when both outputs are the same, the slope is zero and the function is constant.

A linear function $h$ models the depth of a submarine in meters. The function satisfies $h(0) = 450$ and $h(5) = 450$. Which equation defines $h$?

A) $h(x) = 450$
B) $h(x) = 900$
C) $h(x) = 5$
D) $h(x) = 0$

The slope is $m = \dfrac{450 - 450}{5 - 0} = 0$. With $b = 450$, the function is $h(x) = 450$. The answer is A. Don't confuse the slope (0) with the constant value (450).

 

Proportional Relationships from a Table

When a table of values represents a proportional relationship (no y-intercept term), the equation has the form $f(x) = kx$. To find $k$, divide any output by its input.

A scientist measures the mass of different volumes of a liquid at a constant temperature. The table shows the relationship between the volume $v$, in cubic centimeters, and its mass $m(v)$, in grams.

Volume ($v$)	Mass ($m(v)$)
150	375
220	550
300	750

Which equation could define the function $m$?

A) $m(v) = 2.5v$
B) $m(v) = 0.4v$
C) $m(v) = 375v$
D) $m(v) = 225v$

Divide output by input: $k = \dfrac{375}{150} = 2.5$. Check: $\dfrac{550}{220} = 2.5$ and $\dfrac{750}{300} = 2.5$. So $m(v) = 2.5v$. The answer is A.

Option B uses the reciprocal ($\dfrac{150}{375} = 0.4$) — a common trap. Always divide output by input, not the reverse.

 

Table with Non-Zero Starting x (Slope + Back-Solve)

When the table doesn't include $x = 0$, you need two steps: (1) compute the slope from two points, then (2) substitute one point back into $y = mx + b$ to solve for $b$.

The table shows the total cost $C(d)$, in dollars, for renting a car for $d$ days. The relationship is linear. Which function represents this relationship?

Days ($d$)	Cost $C(d)$
1	85
5	265

A) $C(d) = 40d + 45$
B) $C(d) = 180d - 95$
C) $C(d) = -95d + 180$
D) $C(d) = 45d + 40$

Step 1 — slope: $m = \dfrac{265 - 85}{5 - 1} = \dfrac{180}{4} = 45$

Step 2 — y-intercept: Using $(1, 85)$: $85 = 45(1) + b$, so $b = 40$.

The equation is $C(d) = 45d + 40$. The answer is D.

This two-step process — slope first, then back-solve for $b$ — is the core technique for this entire pattern.

 

Finding a Specific Parameter

Medium and Hard questions may ask for just the y-intercept $b$, just the slope $m$, or an expression like $m + b$. The process is the same — find both parameters, then answer what's asked.

The value of a car, $V(a)$, in thousands of dollars, depreciates linearly with its age $a$ in years. The function is $V(a) = ma + b$. The table shows: $(2, 25)$, $(4, 20)$, $(6, 15)$. What is the value of $b$?

A) 25
B) 30
C) $-2.5$
D) 20

Slope: $m = \dfrac{20 - 25}{4 - 2} = \dfrac{-5}{2} = -2.5$

Using $(2, 25)$: $25 = -2.5(2) + b \implies 25 = -5 + b \implies b = 30$.

The answer is B. The y-intercept 30 represents the car's value at age 0 (when new). Don't confuse it with the first table value of 25 — that's the value at age 2, not age 0.

 

Evaluating an Expression with m and b

The hardest version asks for something like $p + q$ or $c - m$ after you've found the slope and intercept.

The value of an investment $V(n)$, in dollars, is modeled by $V(n) = pn + q$, where $p$ and $q$ are constants. The table shows: $(4, 500)$, $(7, 350)$, $(10, 200)$. What is the value of $p + q$?

A) 750
B) 700
C) 650
D) $-750$

Slope: $p = \dfrac{200 - 500}{10 - 4} = \dfrac{-300}{6} = -50$

Using $(4, 500)$: $500 = -50(4) + q \implies 500 = -200 + q \implies q = 700$

So $p + q = -50 + 700 = 650$. The answer is C.

Option B (700) is $q$ alone — the question asks for the sum. Option A (750) comes from $q - p$ instead of $q + p$.

 

Word Problems as Data Points

Some Hard questions disguise the data points in a word problem instead of a table. Extract the two ordered pairs, then proceed as usual.

A tutoring service offers a package deal that costs $300 and includes the first 10 hours. For any time beyond 10 hours, there is an additional hourly charge. The total cost for 25 hours is $675. Which function $C$ gives the total cost for $t$ hours of tutoring, where $t \ge 10$?

A) $C(t) = 25t + 300$
B) $C(t) = 25t + 50$
C) $C(t) = 27t$
D) $C(t) = 27t + 300$

Extract the data points: 10 hours costs $300, so $(10, 300)$. 25 hours costs $675, so $(25, 675)$.

Slope: $m = \dfrac{675 - 300}{25 - 10} = \dfrac{375}{15} = 25$

Using $(10, 300)$: $300 = 25(10) + b \implies 300 = 250 + b \implies b = 50$

So $C(t) = 25t + 50$. The answer is B.

Option A incorrectly uses $300 as the y-intercept — but $300 is the cost at $t = 10$, not $t = 0$. Option C uses the average rate ($675 $\div$ 25 = $27) instead of the marginal rate.

 

SPR: Build the Equation, Then Evaluate

Student-produced response questions often add one more step: after finding the equation, plug in a new value.

The table shows two values for temperature in degrees Zarn (°Z) and their corresponding values in degrees Celsius (°C). There is a linear relationship between the two scales. A temperature reading of $\dfrac{2}{5}$ °Z corresponds to $c$ °C. What is the value of $c$?

°Z	°C
$-8$	$-22$
$12$	$53$

Slope: $m = \dfrac{53 - (-22)}{12 - (-8)} = \dfrac{75}{20} = \dfrac{15}{4}$

Using $(12, 53)$: $53 = \dfrac{15}{4}(12) + b = 45 + b$, so $b = 8$.

The equation is $c = \dfrac{15}{4}z + 8$. Now evaluate at $z = \dfrac{2}{5}$:

$$c = \dfrac{15}{4} \cdot \dfrac{2}{5} + 8 = \dfrac{30}{20} + 8 = 1.5 + 8 = \boldsymbol{9.5}$$

 

Summary of the Process

Every question in this pattern follows the same recipe, regardless of difficulty:

1. Identify two points — from a table, from function values, or from a word problem.
2. Compute the slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.
3. Find the y-intercept: substitute one point into $y = mx + b$ and solve for $b$. (If you already have the point $(0, b)$, you get $b$ immediately.)
4. Write the equation: $f(x) = mx + b$.
5. Answer what's asked — the equation itself, a specific parameter, or an evaluated expression.