Write a function that models the new path of the water

Question

You are watering a garden. The height h (in feet) of water spraying from the garden hose can be modeled by h(x)=-0.2x^2+0.2x+1.2 , where x is the horizontal distance (in feet) from where you are standing. You raise the hose so that the water hits the ground 1 foot farther from where you are standing. Write a function that models the new path of the water.

Answer 1

This is mostly a conceptual question.

When we look at a function h₀(x) = 0, we are looking at where (in this case) the water hits the ground. This is what we are doing when we find the roots of an equation. So the base equation is -0.2x²+0.2x+1.2 = 0.

Let's call the two real roots of this equation a and b. a is behind us let's say so we aren't interested in it. The question is asking what do we need to do to the equation to make b increase by 1.

So then the question becomes how do we manipulate the curve to be translated 1 foot in the positive x direction. The answer is that we look at the properties of quadratic translation.

When we want to translate the graph to the right, we subtract from x and when we want to translate left, we add to x. So in this case we want to go right so we subtract the amount that we want to translate to the right.

Our new equation becomes h₁(x) = -0.2(x-1)²+0.2(x-1)+1.2

We can simplify this into standard form and once we do that, observe what happened to the roots. Both translated 1 unit to the right.

I highly encourage that you play around with this to understand it conceptually and visually before you try to understand its algebraic implications. You can use an online graphing calculator for free like desmos.com to experiment with this.