Question

Equation of the graph: y=-x²+36 , x-intercepts: (-6,0) (6,0) , y-intercept: (0,36)

1. 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.
Analyze the two functions.

Answer the following reflection questions in complete sentences.

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

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

c. 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.

d. Create your own piecewise function with at least two functions. Explain, using complete sentences, the steps for graphing the function. Graph the function by hand or using a graphing software of your choice (remember to submit the graph).

Answer 1

To create a table of values that includes two points of intersection between the airplane and the rainbow, substitute different x values into the equation of the rainbow and solve for y. The domain of the rainbow is all real numbers, and the range is [0, 36]. The x-intercepts represent where the rainbow crosses the x-axis, and the y-intercept represents the highest point of the rainbow on the y-axis. The linear function created from the table can be both positive and negative depending on the slope.

Step-by-step explanation:

To create a table of values that includes two points of intersection between the airplane and the rainbow, we can substitute different x values into the equation of the rainbow and solve for y. Let's choose x = -6 and x = 6, since these are the x-intercepts.

For x = -6, substituting into the equation y = -x^2 + 36 gives us y = -(-6)^2 + 36 = 0.

For x = 6, substituting into the equation gives us y = -(6^2) + 36 = 0.

Therefore, the two points of intersection between the airplane and the rainbow are (-6, 0) and (6, 0).

a. The domain of the rainbow is all real numbers because there are no restrictions on the x-values. The range of the rainbow is [0, 36] because the maximum value of y is 36 and the minimum value is 0.

b. The x-intercepts of the rainbow are (-6, 0) and (6, 0), which represent the points where the rainbow crosses the x-axis. The y-intercept of the rainbow is (0, 36), which represents the highest point of the rainbow on the y-axis.

c. The linear function created with the table can be both positive and negative depending on the slope.

d. I am sorry, but the question doesn't specify what piecewise function to create.