Construct a linear function from a description, two points, a table or a graph, and interpret its rate of change and initial value

Learning goal: Construct a function to model a linear relationship between two quantities. Determine the rate of change and the initial value from a description of the relationship, from two (x, y) values, from a table or from a graph, and interpret both numbers in terms of the situation being modeled and in terms of the graph or table.

The two numbers that fix a linear function. A linear relationship between two quantities is modeled by y = mx + b. Once the rate of change m and the initial value b are known, the whole function is known: every input can be evaluated. Constructing a linear function therefore means finding those two numbers from whatever information is given, and then writing the equation with the variables named for the situation.

Rate of change. The rate of change is how much the output changes for each increase of 1 in the input. Between any two points (x₁, y₁) and (x₂, y₂) on a linear function with different inputs (x₂ ≠ x₁, so that the division is possible), the rate is the change in y divided by the change in x: (y₂ − y₁) ÷ (x₂ − x₁). The order must match: if you subtract the first y from the second y, you must also subtract the first x from the second x. Mixing the orders flips the sign and produces a rate with the wrong direction.

Initial value. The initial value is the output when the input is 0. In the equation it is b. On a graph it is the height where the line crosses the vertical axis. In a table it is the output in the row where x = 0, if that row exists. If the given information does not include x = 0, substitute a known point into y = mx + b and solve for b.

From a description. Suppose a hypothetical pool already contains 300 liters of water and a hose adds 25 liters each minute. Let t be minutes and V be liters in the pool. The rate of change is 25 liters per minute, and the initial value is 300 liters, so V = 25t + 300. After 8 minutes, V = 25 × 8 + 300 = 500 liters. Interpreted in the situation, 25 is the filling speed and 300 is the water present before the hose starts. On the graph, 25 is the rise for each minute to the right and 300 is where the line crosses the vertical axis. In a table with rows every minute, 25 is the difference between consecutive V-values and 300 is the V in the t = 0 row.

From two (x, y) values: a fully worked model. A candle is 20 cm tall after burning for 2 hours and 14 cm tall after burning for 5 hours. Assume it burns at a steady rate. Let t be hours and h be height in centimeters. The two points are (2,20) and (5,14). The rate of change is (14 − 20) ÷ (5 − 2) = (−6) ÷ 3 = −2 centimeters per hour. The sign is negative because the candle gets shorter. To find the initial value, substitute (2,20) into h = −2t + b: 20 = −2 × 2 + b, so 20 = −4 + b and b = 24. The function is h = −2t + 24. Check with the other point: −2 × 5 + 24 = −10 + 24 = 14, which matches. Interpretation: the candle was 24 cm tall before it was lit, and it loses 2 cm of height each hour. It reaches height 0 when −2t + 24 = 0, that is, at t = 12 hours, which is where the graph crosses the horizontal axis. Beyond 12 hours the model no longer describes anything real, because a candle cannot have negative height.

From a table. A table for a hypothetical repair service that charges at a steady hourly rate lists hours t = 3, 6, 9 and total cost C = 41, 62, 83, in dollars. Consecutive rows differ by 3 hours and 21 dollars, so the rate of change is 21 ÷ 3 = 7 dollars per hour. There is no t = 0 row, so substitute (3,41): 41 = 7 × 3 + b gives b = 41 − 21 = 20. The function is C = 7t + 20. Check: 7 × 9 + 20 = 83. In the situation, 7 is the charge per hour of work and 20 is a fixed charge that applies even before any hours are worked, such as a call-out fee. A common error is to call 41 the initial value because it is the first row; but 41 is the cost at t = 3, not at t = 0.

From a graph. Reading two points from a graph works exactly like the two-point method, and the initial value can often be read straight from the vertical axis. For a line through (0,300) and (4,400), the rate is (400 − 300) ÷ 4 = 25 and the initial value is 300, read at the vertical axis. Choose points at grid intersections so the coordinates are exact. If the line crosses the vertical axis between grid lines, estimate and say the initial value is approximate, or find it algebraically from two exactly-read points.

Interpreting the signs and the units. A positive rate of change means the quantity grows as the input grows; a negative rate means it shrinks; a rate of 0 means it stays constant. Always attach units to both numbers: liters per minute, centimeters per hour, dollars per hour for the rate; liters, centimeters, dollars for the initial value. The rate carries a "per" and the initial value does not.

When x = 0 is outside the story. Sometimes the initial value is a mathematical starting point rather than a moment that occurred. If a plant's height is modeled from week 3 onward, the initial value is what the model predicts for week 0, which may or may not match what actually happened before measurements began. State the range of inputs the model is meant to cover, and do not read the initial value as a fact about the world when the situation does not reach back to x = 0.

Worked from two points: A line passes through (1,9) and (4,21). The rate of change is (21 − 9) ÷ (4 − 1) = 12 ÷ 3 = 4. Substitute (1,9): 9 = 4 × 1 + b, so b = 5. The function is y = 4x + 5. Check with (4,21): 4 × 4 + 5 = 21. If the points had been subtracted in mixed order, (21 − 9) ÷ (1 − 4) = −4, the sign would be wrong and the line would appear to fall when it actually rises.

Worked from a table with a negative rate: A linear function's table lists x = 2, 4, 6 and y = 17, 13, 9. The change in y is −4 for each change of 2 in x, so the rate of change is −4 ÷ 2 = −2. Substitute (2,17): 17 = −2 × 2 + b, so b = 21. The function is y = −2x + 21. Check with (6,9): −2 × 6 + 21 = 9. If y were the liters left in a leaking container after x minutes, the container held 21 liters at the start and loses 2 liters per minute.

Worked from a description and interpreted on a graph: A hypothetical taxi charges 3 dollars to start plus 2 dollars per kilometer. With d for kilometers and F for fare in dollars, the rate of change is 2 dollars per kilometer and the initial value is 3 dollars, so F = 2d + 3. A 7-kilometer ride costs 2 × 7 + 3 = 17 dollars. On the graph, the line crosses the vertical axis at 3 and rises 2 for each kilometer to the right; in a table with rows at every kilometer, consecutive fares differ by 2 and the d = 0 row shows 3.

Representation task: For the candle function h = −2t + 24, make a table for t = 0, 2, 4, 6, 8, 10 and 12 hours, plot the seven points on labeled axes, and draw the line. Mark the initial value on the vertical axis and label it with its meaning. Mark the point where the line meets the horizontal axis and explain what it means. Write one sentence interpreting the rate of change in the situation and one interpreting it on the graph.

Independent performance: Choose a real or hypothetical situation with a steady rate of change (a savings balance, a draining tank, a temperature rising evenly, a fee plus a per-unit charge). Give a partner the situation in two of the four forms: a verbal description, two (x, y) values, a table without an x = 0 row, or a graph. The partner must construct the function, state the rate of change and initial value with units, interpret both in the situation and on the graph, evaluate the function at a new input and state the range of inputs for which the model makes sense. Success criteria: the rate is computed as change in y over change in x with matching order; the initial value is found at x = 0, not at the first row; both numbers carry units; the check with a second point is shown; the interpretation says what each number means in the story.