Question

After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. You notice the rainbow is the shape of a parabola. The equation for this parabola is y = -x2 + 36. 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

1) Equation for the parabola y = - x² + 36.


2) 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.


The points of intersection will be: (0,36) and (4, 20)


Table of values for a linear function and the parabola

x        linear function      parabola y = - x² + 36

0        36                                  (0)² + 36 = 36

1        32                                  - (1)² + 36 = 35

2        28                                  - (2)² + 36 = 32

3        24                                   -3² + 36 = 27

4        20                                   -4² + 36 = 20

It is a linear function because the rate of change is constant: for each increase of 1 unit in x, there is a decrease of 4 units in y: ⇒ slope = - 4

Remember to include the two points of intersection in your table. Analyze the two functions; they are there (0,36) and (4, 20)


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.


The domain is all the real values of x between - 6 and 6: - 6 ≤ x ≤ 6


The range is all the values between 0 and 36: 0 ≤ y ≤ 36

The domain represents the horizontal distance between the extremes of the rainbow at the ground.

The range represents: the altitude of the rainbow from the ground (y = 0). The height is at x = 0, y = 36.

The x-intercepts are - 6 and 6, that is the opening of the rainbow at the ground level.


The y-intercept is when x = 0, it is y = 36, and is the height of the rainbow.

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.

The linear function is positive and decreasing. It is positive because it goes above the x-axis, this is y ≥ 0.

The solutions where already signaled: (0,6) and (4, 20). They are the points at which the drone intercepts the rainbow.