Suppose the graph of f(x) = √x is shifted down 4 units and to the left 5 units. What is the equation of the new graph? Verify the result graphically.
The equation of the new graph is g(x) = (Simplify your answer.)
Answer (2)
Shift the original function f(x) = √x down by 4 units: x − 4 .
Shift the result to the left by 5 units: x + 5 − 4 .
The equation of the new graph is: g ( x ) = x + 5 − 4 .
Explanation
Understanding the Problem The problem asks us to find the equation of a new graph, g(x), which is derived from the original function f(x) = √x through two transformations: a vertical shift down by 4 units and a horizontal shift to the left by 5 units. We need to apply these transformations sequentially to obtain the final equation for g(x).
Vertical Shift To shift the graph of f(x) down by 4 units, we subtract 4 from the function. This gives us a new function: f ( x ) − 4 = x − 4
Horizontal Shift To shift the graph to the left by 5 units, we replace x with (x + 5) in the function. Applying this to the result from the previous step, we get: x + 5 − 4
Final Equation Combining both transformations, the equation of the new graph g(x) is: g ( x ) = x + 5 − 4
Examples
Imagine you are designing a game where a character's jumping ability is determined by a square root function. Initially, the jump height is f(x) = √x. To make the game more challenging, you decide to lower the base jump height by 4 units and shift the starting point 5 units to the left. The new jump height function, g(x) = √(x + 5) - 4, now reflects these changes, affecting the character's movement and the game's difficulty. This transformation of functions is a fundamental concept in game development, allowing designers to fine-tune gameplay mechanics.
The new equation after shifting the graph of f(x) = √x down 4 units and left 5 units is g(x) = √(x + 5) - 4. This applies the transformations sequentially, first adjusting for the vertical shift and then for the horizontal shift. The complete transformation results in the final equation for the new graph.
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