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The problem you're working on involves modeling the path of a grasshopper's jump. The given height function for the jump is:

\[
h(x) = \frac{1}{20}x^2 + x
\]

where \( x \) represents the horizontal distance traveled in inches. The problem then mentions that the grasshopper jumps off a rock, and it lands 2 inches farther away. We are tasked with writing a function that models the new path of the jump.

Here’s how to solve it:

1. The original height function is \( h(x) = \frac{1}{20}x^2 + x \). The grasshopper lands 2 inches farther away, so we need to modify this function to reflect that shift.

2. To shift the function horizontally by 2 inches, we replace \( x \) in the original function with \( (x - 2) \). This accounts for the fact that the entire graph is shifted 2 inches to the right.

The new function becomes:

\[
f(x) = \frac{1}{20}(x - 2)^2 + (x - 2)
\]

Now, let's expand this to find the new explicit form of the function:

\[
f(x) = \frac{1}{20}(x^2 - 4x + 4) + (x - 2)
\]

Distribute the \(\frac{1}{20}\) and combine like terms:

\[
f(x) = \frac{1}{20}x^2 - \frac{4}{20}x + \frac{4}{20} + x - 2
\]

Simplify the constants:

\[
f(x) = \frac{1}{20}x^2 - \frac{1}{5}x + \frac{1}{5} + x - 2
\]

Combine the linear terms and constant terms:

\[
f(x) = \frac{1}{20}x^2 + \left(1 - \frac{1}{5}\right)x + \left(\frac{1}{5} - 2\right)
\]

\[
f(x) = \frac{1}{20}x^2 + \frac{4}{5}x - \frac{9}{5}
\]

Thus, the new function that models the grasshopper's path is:

\[
f(x) = \frac{1}{20}x^2 + \frac{4}{5}x - \frac{9}{5}
\]

Would you like more details on any of the steps?

Here are 5 related questions:
1. How do horizontal shifts affect the graph of a quadratic function?
2. What happens to the maximum or minimum point of a quadratic function when it's shifted?
3. Can you explain the effect of vertical shifts on the original jump function?
4. How can you calculate the vertex of a quadratic function like the one in this problem?
5. What is the physical interpretation of the coefficients in the jump function?

**Tip:** When shifting functions, remember that horizontal shifts move the graph left or right, while vertical shifts affect the height without altering the horizontal position.

Modeling the new path of a grasshopper's jump after it lands 2 inches farther away.

Learn how to model the new path of a grasshopper's jump using a quadratic function. This involves shifting the function horizontally and simplifying the result. Suitable for high school students studying algebra and quadratic functions.

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