Interpret intercepts, intervals, extrema, symmetry, end behavior and periodicity of a function that models a situation, and sketch a graph from a verbal description

Learning goal: For a function that models a relationship between two quantities, read the key features of its graph or table and interpret each one in terms of the quantities: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Then reverse the process and sketch a graph showing those key features from a verbal description of the relationship. The focus is on linear, quadratic and exponential functions, with one periodic example so that the word periodicity has a picture attached to it.

Key features are statements about the situation, not just about the picture. A graph of a function that models two quantities is a record of how one quantity depends on the other. Every feature of the picture therefore translates into a sentence about the quantities. An intercept on the vertical axis is the output when the input is zero; an intercept on the horizontal axis is an input that produces output zero. An interval where the function is increasing is a stretch of inputs over which the output goes up as the input goes up. A relative maximum is an output at least as large as all the outputs at nearby inputs (a strict relative maximum is larger than all of them). Reading a graph well means writing these sentences with the real names and units of the quantities, not with the letters x and y.

One model, read completely. A ball is thrown upward from a platform. Its height in meters after t seconds is h(t) = −5t² + 20t + 25, for the time from the throw until the ball reaches the ground. Compute a table: h(0) = 25, h(1) = −5 + 20 + 25 = 40, h(2) = −20 + 40 + 25 = 45, h(3) = −45 + 60 + 25 = 40, h(4) = −80 + 80 + 25 = 25 and h(5) = −125 + 100 + 25 = 0.

The vertical intercept is h(0) = 25: the ball starts 25 meters above the ground, so the platform is 25 meters high. The horizontal intercept is where h(t) = 0. Factoring, −5t² + 20t + 25 = −5(t² − 4t − 5) = −5(t − 5)(t + 1), so h(t) = 0 at t = 5 and at t = −1. Only t = 5 belongs to the situation, because time before the throw is not part of the model; the ball lands 5 seconds after it is thrown. The function is positive on the interval 0 ≤ t < 5, which says the ball is above the ground for the whole flight. The output rises from t = 0 to t = 2 and falls from t = 2 to t = 5, so the function is increasing on 0 < t < 2 and decreasing on 2 < t < 5. The relative maximum is h(2) = 45: the ball reaches its greatest height, 45 meters, 2 seconds after the throw. Because 45 is also the largest value anywhere on the domain, it is the absolute maximum as well. The table shows a symmetry: h(1) = h(3) = 40 and h(0) = h(4) = 25. The parabola is symmetric about the vertical line t = 2, which means the ball takes as long to rise from 40 meters to 45 meters as to fall back from 45 meters to 40 meters. The symmetry belongs to the whole parabola: within the flight it pairs times that both lie in 0 ≤ t ≤ 5, such as 1 and 3, while the landing time t = 5 is paired with t = −1, which is outside the flight. Every quadratic graph has this mirror symmetry about the vertical line through its vertex.

Increasing on an interval, not at a point. A common mistake is to say a function is increasing at t = 1. Increasing is a property of an interval: on 0 < t < 2, any two inputs in that interval have outputs in the same order as the inputs. The turning point itself, t = 2, is where the behavior changes; it belongs to neither the increasing interval nor the decreasing interval in the way the description is usually written. A second mistake is to confuse the interval where the function is increasing with the interval where it is positive. On 2 < t < 5 the ball is still above the ground, so the function is positive there, but it is decreasing. Positive describes the sign of the output; increasing describes the direction of change.

End behavior. End behavior describes what the output does as the input grows without bound in either direction. It is a feature of the whole algebraic function, and it may not belong to the situation. For h(t) = −5t² + 20t + 25 the leading term −5t² dominates for large |t|, so h(t) decreases without bound as t increases and also as t decreases. In the model the domain stops at t = 5, so the end behavior only tells you that a downward parabola cannot describe the ball after it hits the ground. For an exponential model the end behavior is the whole story. A bacteria culture that starts with 200 cells and grows by 50 percent each hour is modeled by P(t) = 200(1.5)ᵗ, with P(0) = 200, P(1) = 300, P(2) = 450, P(3) = 675 and P(4) = 1,012.5. As t increases, P(t) increases without bound; as t decreases toward negative values, P(t) gets closer and closer to 0 but never reaches it. The horizontal line P = 0 is a horizontal asymptote. An exponential growth function has a vertical intercept, here 200, but no horizontal intercept, because a positive number times any power of a positive base is positive, never zero. A linear model has the simplest end behavior of all: a line with positive slope rises without bound as x increases and falls without bound as x decreases, a line with negative slope does the reverse, and a horizontal line stays at one value; unless it is horizontal, a line has exactly one intercept on each axis (a line through the origin has both intercepts at the same point, (0, 0)).

Periodicity. A function is periodic when its outputs repeat at regular intervals of the input: there is a fixed positive number p with f(t + p) = f(t) for every t in the domain (and t + p is in the domain whenever t is). Any such p is a period; the smallest positive one, when one exists, is called the period. Suppose, hypothetically, a table of water depth at a pier records 3.0 meters at midnight, 4.5 meters at 3 a.m., 3.0 meters at 6 a.m., 1.5 meters at 9 a.m., and then 3.0, 4.5, 3.0 and 1.5 meters again at noon, 3 p.m., 6 p.m. and 9 p.m., and suppose the high and low water of each cycle fall at those readings. The pattern repeats every 12 hours, so the period is 12 hours. The relative maximums are 4.5 meters at 3 a.m. and 3 p.m. and the relative minimums are 1.5 meters at 9 a.m. and 9 p.m. This periodic function has infinitely many relative maximums, one in each period, and its end behavior is neither rising nor falling without bound: the outputs stay between 1.5 and 4.5 forever. (Not every periodic function is bounded; the tangent function, met in a later course, repeats yet grows without bound near its asymptotes.)

Sketching from a verbal description. The reverse task is to draw a graph whose key features match a description. Suppose you are told: the temperature of a cup of coffee is 90 °C when it is poured, falls quickly at first and then more slowly, and levels off toward the 20 °C temperature of the room. The sketch needs a vertical intercept at 90, a decreasing curve everywhere, a curve that is steep on the left and flattening to the right, and a horizontal asymptote at 20 that the curve approaches from above. No maximum or minimum appears inside the domain, and the function is never below 20. Every sentence in the description has become a feature of the drawing, and a reader can check the drawing against the sentences one by one.

Worked reading of a table: A hypothetical company records its profit, in thousands of dollars, in each of the years 0 through 6 after opening: −8, −3, 2, 7, 12, 17, 22. The differences between consecutive entries are all 5, so a linear model fits every entry: p(t) = 5t − 8. (The table gives only seven yearly values; treating profit as linear between and beyond them is a modeling assumption.) The vertical intercept −8 means a loss of 8 thousand dollars in the opening year. The horizontal intercept is where 5t − 8 = 0, t = 1.6, so the company crossed from loss to profit about 1.6 years after opening; the function is negative for t < 1.6 and positive for t > 1.6. The function is increasing on the whole domain and has no relative maximum or minimum, which matches a line: it turns nowhere.

Worked reading of a quadratic graph: A rectangular pen is built against a wall with 40 meters of fencing on the other three sides. If the two sides perpendicular to the wall have length x, the parallel side has length 40 − 2x and the area is A(x) = x(40 − 2x) = −2x² + 40x. The horizontal intercepts are x = 0 and x = 20, where the pen has no area; the function is positive on 0 < x < 20, which is the only part of the graph that describes a real pen. By symmetry the vertex is halfway between the intercepts, at x = 10, and A(10) = 10 × 20 = 200. So the area increases on 0 < x < 10, reaches its maximum of 200 square meters when the perpendicular sides are 10 meters, and decreases on 10 < x < 20.

Worked sketch from a description: The number of people in a stadium is zero when the gates open, rises steadily for two hours to 30,000, holds at 30,000 for three hours, then drops rapidly to zero over the next half hour. The sketch has a vertical intercept at 0, a rising segment from (0, 0) to (2, 30000), a horizontal segment from (2, 30000) to (5, 30000) where the function is neither increasing nor decreasing, and a steep falling segment from (5, 30000) to (5.5, 0). The maximum value 30,000 occurs on the whole interval 2 ≤ t ≤ 5 rather than at a single point, and the horizontal intercepts are t = 0 and t = 5.5.

Representation task: For h(t) = −5t² + 20t + 25, make a table for t = 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5 and 5, plot the eleven points, and label on the graph the vertical intercept, the horizontal intercept, the maximum and the axis of symmetry. Under the graph write six sentences, one for each of these features: the intercepts, the interval where the function is increasing, the interval where it is decreasing, the interval where it is positive, the maximum, and the symmetry, each phrased in terms of the ball, meters and seconds. Check that h(1.5) = 43.75 and h(2.5) = 43.75 appear in your table and say what that equality means physically.

Independent performance: Choose a situation of your own that a linear, quadratic or exponential function models, write the function with units, and produce a table of at least eight values and a graph. Then write a paragraph that identifies every key feature that applies: intercepts, intervals of increase and decrease, intervals where the output is positive or negative, relative maximums and minimums, symmetry, end behavior and whether the function is periodic. Finally, write a verbal description of a different relationship and sketch its graph from the description alone. Success criteria: every value in the table is correct; each key feature is interpreted as a sentence about the quantities, with units; features that do not exist (for example a maximum of a linear function) are explicitly said not to exist, with a reason; and the sketch from the description has each stated feature visibly present.