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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.

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Beanie4185

Remember that a quadratic with two real zeroes can be written as \(a(x - r_1)(x - r_2)\), where \(a\) is a constant and \(r_1\) and \(r_2\) 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

\(y = a(x - ({-6}))(x - 6)\), or
\(y = a(x + 6)(x - 6)\)

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

\(36 = a(0 + 6)(0 - 6)\)
\(36 = {-36}a\)
\(a = {-1}\)
(It makes sense that \(a\) is negative since the parabola opens down.)

So, the equation of the parabola is

\(y = -(x + 6)(x - 6)\), or
\(\bf y = -x^2 + 36\)

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:

\(y = -({-2})^2 + 36 = {-4} + 36 = 32\)
\(y = -(1)^2 + 36 = {-1} + 36 = 35\)

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 \(f(x) = mx + b\). The slope \(m\) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\), so

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{35 - 32}{1 - ({-2})} = 1\)
So, the equation of the linear function so far is just \(f(x) = x + b\), and we can find \(b\) by plugging in one of the points on the line:

\(35 = 1 + b\)
\(b = 34\)

Thus, the equation of the linear function is

\(\bf f(x) = x + 34\)

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

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