please help with this algebra 2 word problem

Question

The function h(x)=−0.03(x−14)2+6 models the jump of a red kangaroo, where x is the horizontal distance traveled (in feet) and h(x) is the height (in feet). When the kangaroo jumps from a higher location, it lands 5 feet farther away. Write a function that models the second jump. Round to the nearest hundredths.

Answer 1

h(x) = -0.03(x − 14ft)² + 6ft is the original equation. Let's plot a few points to understand the motion of the kanagaroo's jump:

 

Note: The notation below is read "a-not", "b-not", and "c-not". They are used to mark original points. Later, we will use "a-one", "b-one", and "c-one" to show change. Always feel free to change notation to whatever's easiest for you. If you'd like, you can use a, b, and c now, and d, e, and f later.

 

a₀) at a horizontal distance of 0ft, the kangaroo is at .12ft high, or about 1.5 inches.

b₀) at a horizontal distance of 14ft, the kangaroo is at it's maximum height (vertex) of 6ft.

c₀) at a horizontal distance of 28ft, the kangaroo has returned to it's original height of .12ft.

 

So, the kangaroo "lands" at .12ft.

 

Next, think about the changes to the graph/equation. The initial height a₀ is being moved up an unknown number of units, and it's causing the landing c₀ to occur 5ft later, at 33ft. We need to know how much higher to shift our parabola to effect the intercept in such a way. Our new landing, c₁, occurs at (33 , 0.12).

 

Since the only shift that occurs to the graph is a vertical one, the vertex of the parabola will stay at a horizontal distance of 14ft. The only piece of the original equation h(x) = -0.03(x − 14ft)² + 6ft that will change is the "6", or the k in the vertex. Now we have all of the information we need to solve for the new k.

 

c₁ = (x₁ , y₁) = (33 , 0.12)

a = -0.03                                   **unchanged from original equation

b₁ = (h₁ , k₁) = (14, k₁)

 

y₁ = a(x₁ − h₁)² + k₁             plug in

0.12 = -0.03(33 − 14)² + k₁              simplify

0.12 = -0.03(19)² + k₁

0.12 = -0.03(361) + k₁

0.12 = -10.83 +k₁

10.95 = k₁

 

so our new vertex b₁ = (14 , 10.95)

 

Finally, or new equation for the second jump of the kangaroo is

 

y = a(x − h)² + k

h(x) = 0.03(x − 14ft)² + 10.95ft