Question

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.

Answer 1

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).