Question

After a dreary day of rain, the sun peeks through the clouds and rainbow forms. You notice the rainbow is the shape of a parabola.
The equation for this parabola is y = -x^2 + 36.

Graph of a parabola opening down at the vertex 0 commas 36 crossing the x–axis at negative 6 commas 0 and 6 commas 0.

Create a table of values for a linear function. A drone is in the distance, flying upward in a straight line. It intersects the rainbow at two points. Choose the points where your drone intersects the parabola and create a table of at least four values for the function. Remember to include the two points of intersection in your table.

Analyze the two functions. Answer the following reflection questions in complete sentences.

What is the domain and range of the rainbow? Explain what the domain and range represent. Do all of the values make sense in this situation? Why or why not?
What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.
Is the linear function you created positive or negative? Explain.
What are the solutions or solution to the system of equations created? Explain what it or they represent.

Answer 1

Answer:

We are given that,

The equation of the rainbow is represented by the parabola,

$y=-x^2+36$

Now, we are required to find a linear equation which cuts the graph of the parabola at two points.

Let us consider the equation joining the points (-6,0) and (0,36), given by $y=6x+36$.

So, the corresponding table for the linear equation is given by,

x             $y=6x+36$

-6                      0

0                       36

1                        42

6                       72

Now, we will answer the questions corresponding the functions.

1. Domain and Range of the rainbow.

Since, the equation of the rainbow is $y=-x^2+36$

So, from the figure, we get that,

Domain is the set of all real numbers.

Range is the set $\{ y|y\leq 36 \}$

Here, domain represents the points which are used to plot the path of the rainbow and range represents the points which are form the rainbow.

Not all points make sense in the range as the parabola is opening downwards having maximum point as (0,36).

2. X and Y-intercepts of the rainbow.

As, the 'x and y-intercepts are the points where the graph of the function cuts x-axis and y-axis respectively i.e. where y=0 and x=0 receptively'.

We see that from the figure below,

X-intercepts are (-6,0) and (6,0) and the Y-intercept is (0,36)

Here, these intercepts represents the point where the parabola intersects the individual axis.

3. Is the linear function positive or negative.

As the linear function is $y=6x+36$ represented by the upward flight of the drone.

So, the linear function is a positive function.

4. The solution of the system of equations is the intersection points of their graphs.

So, from the figure, we see that the equations intersect at the points (-6,0) and (0,36).

Thus, the solution represents the position when both the drone and rainbow intersect each other.

Answer 2

In the picture attached, both the plot of the parabola and of the straight line are shown. The parabola represents the rainbow and the line, the path of the drone.  

Drone path

x y

-5 11

-4 15

0 31

1 35

What is the domain and range of the rainbow? Explain what the domain and range represent.  

The domain of a parabola is all real numbers, the range of this parabola is [-infinite, 36].   The domain represents all possible values x-variable can take. The range represents all possible values y-variable can take.

Do all of the values make sense in this situation? Why or why not?

No, it doesn't. Only values between [0, 36] of the range make sense, because negative values are below the horizon, where there is no rainbow

What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.

x-intercepts: (-6, 0) and (0, 6). They represent the points at which the rainbow intercept the horizon.

y-intercepts: (0, 36). It represents the maximum height of the rainbow.

Is the linear function you created positive or negative? Explain.

The linear function created has a positive slope. As a consequence, the function is always increasing.

What are the solutions or solution to the system of equations created? Explain what it or they represent.

The solutions are the points at which the parabola and the line intercept. In this case they are points (-5, 11) and (1,35)