Math Problem Statement

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

Solution

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)=120x2+xh(x) = \frac{1}{20}x^2 + x

where xx 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)=120x2+xh(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 xx in the original function with (x2)(x - 2). This accounts for the fact that the entire graph is shifted 2 inches to the right.

The new function becomes:

f(x)=120(x2)2+(x2)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)=120(x24x+4)+(x2)f(x) = \frac{1}{20}(x^2 - 4x + 4) + (x - 2)

Distribute the 120\frac{1}{20} and combine like terms:

f(x)=120x2420x+420+x2f(x) = \frac{1}{20}x^2 - \frac{4}{20}x + \frac{4}{20} + x - 2

Simplify the constants:

f(x)=120x215x+15+x2f(x) = \frac{1}{20}x^2 - \frac{1}{5}x + \frac{1}{5} + x - 2

Combine the linear terms and constant terms:

f(x)=120x2+(115)x+(152)f(x) = \frac{1}{20}x^2 + \left(1 - \frac{1}{5}\right)x + \left(\frac{1}{5} - 2\right)

f(x)=120x2+45x95f(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)=120x2+45x95f(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.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Function Shifting
Algebra

Formulas

Original function: h(x) = (1/20)x^2 + x
Shifted function: f(x) = (1/20)(x - 2)^2 + (x - 2)

Theorems

Horizontal Shifting of Functions
Combining Like Terms

Suitable Grade Level

Grades 9-12