Answer
The equation of the yellow graph is $$y = -(x+7)^2$$
The equation of the red graph is: $$y = -(x-4)^2$$
Work Step by Step
The blue graph is the graph of function $y = -x^2$. It collides with the x-axis at point $x = 0$.
1) First, we look at the yellow graph:
- It is a horizontal shift. So the equation of the function of yellow graph has this form:
$$y = -(x + h)^2$$
- The yellow graph shifts to the left of the blue graph, so $h\gt0$.
- The yellow graph collides with the x-axis at point $x=-7$, which is $7$ units away from the blue graph, so $|h|=7$.
- Since we know that $h\gt0$, we figure that $h=7$
Therefore, the equation of the yellow graph is: $$y = -(x+7)^2$$
2) Now, about the red graph:
- It is also a horizontal shift. So the equation of the function of the red graph also has this form:
$$y = -(x + h)^2$$
- However, since the red graph shifts to the right of the blue graph, we figure that $h\lt0$.
- The red graph collides with the x-axis at point $x=4$, which is $4$ units away from the blue graph, so $|h|=4$.
- Yet, because $h\lt0$, we conclude that $h=-4$
Therefore, the equation of the red graph is: $$y = -(x-4)^2$$
