Create an equation. use the graph below to create the equation of the rainbow parabola.
graph of a parabola opening down at the vertex 0 comma 36 crossing the x–axis at negative 6 comma 0 and 6 comma 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.

Remember that a quadratic with two real zeroes can be written as , where is a constant and and are the zeroes (or roots) of the function. Since the graph shows that the two zeroes are at -6 and 6, the equation has to be of the form

, or


To solve for a, let's use the point at the vertex (0, 36) and plug that in:




(It makes sense that is negative since the parabola opens down.)

So, the equation of the parabola is

, or


Now for the second part, just pick any two points with which we can draw a line with a positive slope. I'll use x = -2 and 1:




So, our two points are (-2, 32) and (1, 35). To find the equation of the linear function that goes through these two points, let's use slope-intercept form, which is . The slope is given by , so


So, the equation of the linear function so far is just , and we can find by plugging in one of the points on the line:




Thus, the equation of the linear function is



And you can find more points on the line simply by plugging other values of x, such as (0, 34) and (5, 39).