Question

I know it's a long question but the reward is great!

Please answer if you truly know how to help!! Thank you!!


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.


In the distance, an airplane is taking off. As it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. Create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow.




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 with your table positive or negative? Explain.


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


Again thank you if you can help me with this!!

Answer 1

Explanation:

Here y = -x^2 + 36.

We can choose x values pretty much at random and then calculate the associated y values:

x y = -x^2 + 36

---- --------------------

0 36

2 -2^2 + 36 = 32

-2 -4 + 36 = 32

3 -9 + 36 = 27

You did not share the equation of the slanted line. Let's assume that the equation of this line is y = mx + b. The first two points in the table above are (0, 36) and (2, 32). We could find the equation of this line as follows:

Slope: as we move from (0, 36) to (2, 32), x increases by 2 and y decreases by 4. Thus, the slope is m = rise / run = -4/2, or -2. Using info from the point (0, 36), we find the y-intercept of this straight line:

y = mx + b becomes 36 = m(0) + b, so b = 36, and the line is y = -2x + 36.

We need to find the points of intersection of y = -2x + 36 and y = -x^2 + 36. We can equate these equations to eliminate y: -2x + 36 = -x^2 + 36, or

-2x = -x^2. Equivalently, 2x - x^2 = 0, or (x)(2 - x) = 0. Then x = 0 and x = 2.

This says that the line and the parabola intersect in at least two places:

(0, 36) and (2, 32).