Unit 5 Functions And Volume — Unit Plan

Let’s say we have an input-output rule that gives exactly one output for each allowable input. Then we say the output depends on the input, or the output is a function of the input.

For example, the area of a square is a function of the side length because the area can be found from the side length by squaring it. So when the input is 10 cm, the output is 100 cm².

Sometimes we might have two different rules that describe the same function. As long as we always get the same single output from any given input, the rules describe the same function.

We can sometimes represent functions with equations. For example, the area, $A$, of a circle is a function of the radius, $r$, and we can express this with this equation: $\displaystyle A=\pi r^2$

We can also draw a diagram to represent this function:

In this case, we think of the radius, $r$, as the input and the area of the circle, $A$, as the output. For example, if the input is a radius of 10 cm, then the output is an area of $100\pi$ cm², or about 314 cm². Because this is a function, we can find the area, $A$, for any given radius, $r$.

Since $r$ is the input, we say that it is the independent variable, and since $A$ is the output, we say that it is the dependent variable.

We sometimes get to choose which variable is the independent variable in the equation. For example, if we know that

$\displaystyle 10A-4B=120$

then we can think of $A$ as a function of $B$ and write

$\displaystyle A=0.4B+12$

or we can think of $B$ as a function of $A$ and write

$\displaystyle B=2.5A-30$

Here is the graph showing Noah's run.

A graph in the coordinate plane, horizontal, distance in meters, 0 to 24 by threes, vertical, time in seconds, 0 to 10 by ones. The graph begins at the origin and steadily increases as it moves right, passing through the labeled point at ( 18 comma 6 ).

The time in seconds since he started running is a function of the distance he has run. The point $(18,6)$ on the graph tells us that the time it takes him to run 18 meters is 6 seconds. The input is 18 and the output is 6.

The graph of a function is all the coordinate pairs, (input, output), plotted in the coordinate plane. By convention, we always put the input first, which means that the inputs are represented on the horizontal axis, and the outputs are represented on the vertical axis.

Here is a graph showing the temperature in a town as a function of hours after 8:00 p.m.

The graph of a curve on the coordinate plane. The horizontal axis is labeled “time in hours after 8 pm” and the numbers 1 through 11 are indicated. The vertical axis is labeled “temperature in degrees Fahrenheit” and the numbers 45 through 60, in increments of 3 are indicated. The curve starts on the vertical axis at the point 0 comma 60, and moves downwards and to the right. It continues downward until reaching a minimum point of 8 comma 45, turns, and then moves upward and to the right, passing through the point 11 comma 57.

The graph of a function tells us what is happening in the context the function represents. In this example, the temperature starts out at $60^\circ$ F at 8:00 p.m. It decreases during the night, reaching its lowest point about 8 hours after 8:00 p.m., or 4:00 a.m. Then it starts to increase again.

Here is a graph showing Andre's distance as a function of time.

For a graph representing a context, it is important to specify the quantities represented on each axis. For example, if this is showing distance from home, then Andre starts at some distance from home (maybe at his friend’s house), moves farther away (maybe to a park), then returns home.

If instead the graph is showing distance from school, the story may be Andre starts out at home, moves farther away (maybe to a friend's house), then goes to school.

What could the story be if the graph is showing distance from a park?

Functions are all about getting outputs from inputs. For each way of representing a function—equation, graph, table, or verbal description—we can determine the output for a given input.

Let’s say we have a function represented by the equation $y = 3x +2$, where $y$ is the dependent variable and $x$ is the independent variable. If we wanted to find the output that goes with 2, we could input 2 into the equation for $x$ and find the corresponding value of $y$. In this case, when $x$ is 2, $y$ is 8 since $3\boldcdot 2 + 2=8$.

If we had a graph of this function instead, then the coordinates of points on the graph would be the input-output pairs.

So we would read the $y$-coordinate of the point on the graph that corresponds to a value of 2 for $x$. Looking at the following graph of a function, we can see the point $(2,8)$ on it, so the output is 8 when the input is 2.

The graph of a line in the coordinate plane with the origin labeled “O”. The horizontal axis has the numbers negative 1 through 2 indicated and there are vertical gridlines between each integer. The vertical axis has the numbers negative 2 through 8, in increments of 2, indicated, and there are horizontal grid lines in between each integer. The line begins to the right of the y axis and below the x axis. It slants upward and to the right passing through the point with coordinates negative 1 comma negative 1, crosses the y axis at 2, and passes through the indicated point labeled 2 comma 8.

Suppose a car is traveling at 30 miles per hour. The relationship between the time in hours and the distance in miles is a proportional relationship.

More generally, if we represent a linear function with an equation like $y = mx + b$, then $b$ is the initial value (which is 0 for proportional relationships), and $m$ is the rate of change of the function.

If we represent a linear function in a different way, say with a graph, we can use what we know about graphs of lines to find the $m$ and $b$ values and, if needed, write an equation.

Water has different boiling points at different elevations. At 0 m above sea level, the boiling point is $100^\circ$ C. At 2,500 m above sea level, the boiling point is $91.3^\circ$ C. If we assume the boiling point of water is a linear function of elevation, we can use these two data points to calculate the slope of the line: $\displaystyle m=\frac{91.3-100}{2,500-0}=\frac{\text-8.7}{2,500}$

This slope means that for each increase of 2,500 m, the boiling point of water decreases by $8.7^\circ$ C.

Next, we already know the $y$-intercept is $100^\circ$ C from the first point, so a linear equation representing the data is $\displaystyle y=\frac{\text-8.7}{2,500}x+100$

This equation is an example of a mathematical model. A mathematical model is a mathematical object, like an equation, a function, or a geometric figure, that we use to represent a real-life situation. Sometimes a situation can be modeled by a linear function. We have to analyze the information we are given and use judgment about whether using a linear model is a reasonable thing to do. We must also be aware that the model may make imprecise predictions or may only be appropriate for certain ranges of values.

Testing our model for the boiling point of water, it accurately predicts that at an elevation of 1,000 m above sea level (when $x=1,000$), water will boil at $96.5^\circ$ C (since $y=\frac{\text-8.7}{2,500}\boldcdot 1000+100=96.5$). For higher elevations, the model is not as accurate, but it is still close. At 5,000 m above sea level, it predicts $82.6^\circ$ C, which is $0.6^\circ$ C off the actual value of $83.2^\circ$ C. At 9,000 m above sea level, it predicts $68.7^\circ$ C, which is about $3^\circ$ C less than the actual value of $71.5^\circ$ C. The model continues to be less accurate at even higher elevations since the relationship between the boiling point of water and elevation isn’t linear, but for the elevations in which most people live, it’s pretty good.

This graph shows Andre biking to his friend’s house, where he hangs out for a while. Then they bike together to the store to buy some groceries before racing back to Andre’s house for a movie night. Each line segment in the graph represents a different part of Andre’s travels.

Graph composed of 5 linear sections. Horizontal axis, time, vertical axis, distance from home. Beginning at the origin, first segment slopes up as it moves right. Second segment horizontal, third segment slopes up and right, more steeply than the first segment. Fourth segment horizontal, fifth segment slopes down and right back to the horizontal axis.

This is an example of a piecewise linear function, which is a function whose graph is pieced together out of line segments. It can be used to model situations in which a quantity changes at a constant rate for a while, then switches to a different constant rate.

We can use piecewise functions to represent stories, or we can use them to model actual data. In the second example, temperatures recorded at different times throughout a day are modeled with a piecewise function made up of two line segments. Which line segment do you think does the best job of modeling the data?

Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.