Interpreting a Functional Relationship between Two Quantities

7.14.2 Interpreting a Functional Relationship between Two Quantities

7.14.2 • Interpreting a Functional Relationship between Two Quantities

Activity

The equation $h (t) = 2 + 23.7 t - 4.9 t^{2}$ represents the height of a pumpkin that is catapulted up in the air as a function of time, $t$ , in seconds. The height is measured in meters above ground. The pumpkin is shot up at a vertical velocity of 23.7 meters per second.

1. What do you think the 2 in the equation $h (t) = 2 + 23.7 t - 4.9 t^{2}$ tells us in this situation?

Compare your answer:

The 2 tells us the pumpkin is 2 meters above ground before it is launched.

2. What do you think the $- 4.9 t^{2}$ in the equation $h (t) = 2 + 23.7 t - 4.9 t^{2}$ tells us in this situation?

Compare your answer:

Answer: The $- 4.9 t^{2}$ shows the height lost due to gravity (like the $- 16 t^{2}$ in the cannonball problem).

3. If we graph the equation, will the graph open upward or downward? Why?

The graph would open downward. The height starts at 2 meters and then increases as the pumpkin is shot up, and eventually, it decreases as the pumpkin falls back to the ground

4. Think about where the vertical intercept would be. What is the $y$ -value of that vertical intercept?

The vertical intercept would be at $(0, 2)$ , because when $t$ is 0, the height is 2 meters.

5. What about the horizontal intercepts?

Compare your answer:

The horizontal intercepts are hard to tell without graphing or having the equation in factored form. One of them will be located around $(5, 0)$ because substituting 5 for $t$ gives a height of -2 meters, which is pretty close to 0.

6. Use the graphing tool or technology outside the course. Graph the equation $h (t) = 2 + 23.7 t - 4.9 t^{2}$ .

Access multimedia content

Compare your answer:

A parabola on a coordinate grid. The x-axis represents the time in seconds and the y-axis represents the height in meters. The x-axis scale is 0.5 and extends from 0 to 6. The y-axis scale is 4 and extends from 0 to 36.

7. Now that you have graphed the equation, answer questions a–c using the graph.

a. What is the vertical intercept and what does this point mean in our catapulted pumpkin situation?

Compare your answer:

The vertical intercept is $(0, 2)$ . It tells us that the pumpkin's initial position is 2 meters above ground.

b. What are the horizontal intercepts and what do these points mean in our catapulted pumpkin situation?

Compare your answer:

The graph intersects the horizontal axis around -0.08 and 4.9. The positive horizontal intercept tells us that the pumpkin hits the ground after about 4.9 seconds. The negative intercept doesn't have any meaning here since the time cannot be negative in this case.

c. What is the vertex of the parabola and what does this point mean in our catapulted pumpkin situation?

Compare your answer:

The vertex is at approximately $(2.4, 30.7)$ . It tells us that the pumpkin reaches its maximum height of about 30.7 meters around 2.4 seconds after being fired.

Are you ready for more?

Extending Your Thinking

What approximate vertical velocity would this pumpkin need for it to stay in the air for about 10 seconds? (Assume that it is still shot from 2 meters above the ground and that the effect of gravity pulling it down is the same.)

Compare your answer:

If $h (t) = 2 + b t - 4.9 t^{2}$ takes the value 0 when $t$ is close to 10, this means that $2 + 10 b - 490$ is close to zero. That means we want $b$ to be 48.8. The pumpkin needs to be shot in the air at 48.8 meters per second.

Self Check

The function that gives the height, $h$ , in feet of a cannonball $t$ seconds after the ball leaves the cannon is graphed below.

$Graph with time in seconds on $x$-axis and distance above ground in feet on $y$-axis.$

What is the point $(10, 1,600)$ and what does it mean in this situation?

$Y$ -intercept, the height the cannonball was launched
Maximum height, the highest height the cannonball reaches
Maximum height, the height the cannonball was launched
$Y$ -intercept, the highest height the cannonball reaches

Additional Resources

The Meaning of Values on a Graph

Let’s say a tennis ball is hit straight up in the air, and its height in feet above the ground is modeled by the equation $f (t) = 4 + 12 t - 16 t^{2}$ where $t$ represents the time in seconds after the ball is hit. Here is a graph that represents the function, from the time the tennis ball was hit until the time it reached the ground.

A parabola on a coordinate grid. The x-axis represents the time in seconds and the y-axis represents the height in feet. The x-axis scale is 0.25 and extends from 0 to 1.25. The y-axis scale is 1 and extends from 0 to 8.

In the graph, we can see some information we already know, and some new information:

The 4 in the equation means the graph of the function intersects the vertical axis at 4. It shows that the tennis ball was 4 feet off the ground at $t = 0$ , when it was hit.
The horizontal intercept is $(1, 0)$ . It tells us that the tennis ball hit the ground 1 second after it was hit.
The vertex of the graph is at approximately $(0.4, 6.3)$ . This means that about 0.4 second after the ball was hit, it reached the maximum height of about 6.3 feet.

The equation can be written in factored form as $f (t) = (- 16 t - 4) (t - 1)$ . From this form, we can see that the zeros of the function are $t = 1$ and $t = \frac{1}{4}$ . The negative zero, -14, is not meaningful in this situation because the time before the ball was hit is irrelevant.

Try it

The Meaning of Values on a Graph

An object is thrown upward from a height of 5 feet with a velocity of 60 feet per second. Its height h(t) in feet after $t$ seconds is modeled by the function $h (t) = 5 + 60 t - 16 t^{2}$ and graphed below.

What is the value of the vertical intercept ( $y$ -intercept) and what does it mean in the function?

A parabola on a coordinate grid. The x-axis represents the time in seconds and the y-axis represents the distance above the ground in feet. The x-axis scale is 0.5 and extends from 0 to 4.5. The y-axis scale is 20 and extends from 0 to 80.

Here is how to determine the $y$ -intercept:

The vertical intercept, or $y$ -intercept, is where the graph crosses the $y$ -axis. It occurs when $x = 0$ . You can also identify the $y$ -intercept by looking at the constant in the function. This occurs at $y = 5$ . The point is (0,5). The $y$ -intercept represents the initial height. It means the object is thrown from 5 feet above the ground.