A tank holds 600 liters of water and drains at a steady 40 liters per minute. After minutes it holds liters. The two numbers in that formula answer the first two questions anyone asks, how much was there at the start and how fast it is draining. The point where the graph meets the time axis answers the next one, when the tank is empty. Many SAT questions about linear models ask you to say in words what one of these numbers means.
See also Drawing Lines from Slope and Intercepts in Foundations for Calculus and Linear Functions and Constant Rate of Change in Precalculus.
Key terms
Slope-intercept form
Definition Every line that is not vertical has exactly one equation of the form
where is the slope of the line and is the point where it crosses the -axis. A vertical line has no equation of this form.
In plain words Solve for , and the equation shows the slope and the starting height at once.
Example Rewrite . Subtracting gives , and dividing by gives
The slope is and the line crosses the -axis at .
The y-intercept
Definition A -intercept of a graph is a point where the graph meets the -axis. Such a point has -coordinate , so it is found by setting . The graph of a function has at most one -intercept, and it has one exactly when is in the domain. For the line it is the point . Many questions also call the number the -intercept, so read carefully whether a question asks for the point or for its -coordinate.
In plain words The -intercept is where the graph crosses the vertical axis, the output when the input is zero.
Example For , setting gives . The -intercept is the point . A question that asks for the -coordinate of the -intercept wants , and one that asks which point is the -intercept wants .
The x-intercept
Definition An -intercept of a graph is a point where the graph meets the -axis. Such a point has -coordinate , so it is found by setting . A line with has exactly one -intercept, at
A horizontal line with has no -intercept, and the line is the -axis itself.
In plain words The -intercept is where the graph touches the horizontal axis, the input that makes the output zero.
Example For , setting gives , so and the -intercept is . The formula agrees, since
Initial value
Definition In a model whose input measures time from a chosen starting moment, the initial value is the output at input . For a model the initial value is , the -coordinate of the -intercept. What it means depends on what the input stands for. If counts years after 2010, the initial value is the value in 2010, not in the year 0.
In plain words The initial value is where the quantity stands at the moment you start the clock. Move the start, and the initial value moves with it.
Example A town’s population is modeled by , where is the number of years after 2010. The initial value is the population in 2010, and in 2018 the model gives . Now count years after 2015 instead, with . Then
The slope is still people per year, but the initial value is now , the population in 2015. Both models give for 2018, since there.
Linear model
Definition A linear model describes a quantity that depends on a quantity by on a stated set of inputs. The slope is the constant rate of change, in units of per unit of , and is the value of when . The line of the equation extends forever in both directions, but the model is valid only on the inputs the situation allows, its contextual domain. Every statement about slope or intercepts must be read with the axis labels and units.
In plain words A linear model is a line used to describe something real. Its slope is the rate, its intercepts are special moments, and outside the sensible inputs the line keeps going but the story stops.
Example For the tank, with in liters and in minutes.
| Part of the model | Value | Meaning |
|---|---|---|
| slope | the tank loses liters each minute | |
| -intercept | it holds liters at the start | |
| -intercept | it is empty after minutes |
The -intercept comes from , which gives . The model makes sense for . At the formula gives liters, a point on the line that describes nothing real.
Check yourself
- A gym charges a one-time joining fee plus a fixed monthly fee. The total cost, in dollars, of a membership for months is . What is the best interpretation of in this context? (A) The monthly fee is dollars. (B) The joining fee is dollars. (C) The total cost after one month is dollars. (D) The cost increases by dollars each month.
Show the answer
The correct choice is (B). The number is the value of when , the cost before any month has been paid, which is the joining fee. The monthly fee is the slope , so (A) and (D) describe the wrong number. The total after one month is dollars, so (C) fails.
- The volume of water in a tank minutes after it starts draining is liters. How many minutes after it starts draining is the tank empty?
Show the answer
The answer is . The tank is empty when , so and . This is the -coordinate of the -intercept .
- What is the -intercept of the graph of in the -plane? (A) (B) (C) (D)
Show the answer
The correct choice is (C). Setting gives , so and the point is . Choice (A) is the -intercept, found by setting , which gives and . Choice (B) puts that on the wrong axis, and (D) puts the on the wrong axis.