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.

1. the equation for this parabola is y = -x2 + 36.

2. 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.
3. 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.

1 For this question, the points of intersection that I chose for this problem were (-4, 20) and (2, 32). Keep in mind that the linear function can have any two points that cross the parabola but the right point HAS to be lower than the left. This is shown by the question "flying upwards in a straight line." Anyway, now that you have two points, you need to find the y-intercept of the line and the slope. For the points that I have chosen my y-intercept was (0, 28) and the slope was 2/1. Now, make the equation. y = 2x + 28 (remember the graph goes up at intervals of 2) Now, you need to make a table of values. For mine, the x values increased by 1, starting at -4, and the y values increased by 2, starting at 20. That should answer the first part of the question. The second part was a little tougher. For the first bullet point, the domain (x) is all real numbers because the graph expands infinitely. The range (y) is any number less than 36 because 36 is the graph's highest point and it continually expands downward. For the second bullet point, the x-intercepts (you should be able to identify those) are where the rainbow touches the horizon. The y-intercept is where the rainbow reaches its highest point. For the third bullet point, it depends on your linear function. Mine happens to be positive because as the x-values increase, so do the y. If it were negative, the y-values would decrease as the x-increase. For the final bullet point, the solutions of the functions would be where your points of intersection are. To show this, plug the values into both equations and find that you will get the same numbers each time. Hope this helps

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