The equation for the parabola is y=-x^2+36 graph of a parabola opening down at the vertex 0,36 crossing the x-axis at -6,0 and 6,0 read more in the description.

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, 36 crossing the x–axis at negative 6, 0 and 6, 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

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?

Domain: any x from -6 to 6; any absolute value > 6 is not meaningful in this case because the x-axis represents the ground level, and no y values can be negative (underground).

Range: y values from 0 (ground) up to 36 (an upward measure, in some kind of units).

Drone flying upward is y = mx+b; choose m > 0 and b > 0 (e.g. m = 0.5, b = 24)--this will intersect the rainbow twice; you can find those points of intersection by setting the y values of line & parabola =, and solving the resulting quadratic equation by using the quadratic formula.  x values of intersection are about -3.60 and 3.35.