Math Problem Statement
Solution
The problem you're working on involves modeling the path of a grasshopper's jump. The given height function for the jump is:
where 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:
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The original height function is . The grasshopper lands 2 inches farther away, so we need to modify this function to reflect that shift.
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To shift the function horizontally by 2 inches, we replace in the original function with . This accounts for the fact that the entire graph is shifted 2 inches to the right.
The new function becomes:
Now, let's expand this to find the new explicit form of the function:
Distribute the and combine like terms:
Simplify the constants:
Combine the linear terms and constant terms:
Thus, the new function that models the grasshopper's path is:
Would you like more details on any of the steps?
Here are 5 related questions:
- How do horizontal shifts affect the graph of a quadratic function?
- What happens to the maximum or minimum point of a quadratic function when it's shifted?
- Can you explain the effect of vertical shifts on the original jump function?
- How can you calculate the vertex of a quadratic function like the one in this problem?
- 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