Problem 43 - Calculus Single Variable

Question

Two functions \(f\) and \(g\) are given. Find a constant \(h\) such that \(g(x)=f(x+h)\). What horizontal translation of the graph of \(f\) results in the graph of \(g\) ? \(f(x)=\sqrt{1-x^{2}}, g(x)=\sqrt{2 x-x^{2}}\)

Step-by-Step Solution

Answer

The horizontal translation required is 1 unit to the right.

Step 1: Understand the Problem

You are given two functions, \( f(x) = \sqrt{1 - x^2} \) and \( g(x) = \sqrt{2x - x^2} \). You need to find a value \( h \) such that \( g(x) = f(x+h) \). This involves finding what horizontal shift \( h \) needs to be applied to \( f \) to obtain \( g \).

Step 2: Set up the Equation

The equation to solve is \( g(x) = f(x + h) \). Substituting the functions gives \[ \sqrt{2x - x^2} = \sqrt{1 - (x+h)^2}. \] This equation needs to be solved for \( h \).

Step 3: Simplify the Equation

First, eliminate the square roots by squaring both sides: \[ 2x - x^2 = 1 - (x+h)^2. \] This results in the equation: \[ 2x - x^2 = 1 - (x^2 + 2xh + h^2). \] Simplify the right side to get: \[ 2x - x^2 = 1 - x^2 - 2xh - h^2. \]

Step 4: Solve for h

Reorganize the equation to isolate terms involving \( h \): \[ 2x - x^2 = 1 - x^2 - 2xh - h^2. \] This simplifies to \[ 2x = 1 - 2xh - h^2. \] Solving for \( h \) requires adjusting terms: \[ 2xh + h^2 = 1 - 2x. \] Assume \( h \) needs to be constant irrespective of \( x \), implying \( h = 1 \).

Step 5: Verify the Solution

Check if \( h = 1 \) solves the equation. Substitute \( h = 1 \) into \( g(x) = f(x+1) \): \[ \sqrt{2x - x^2} = \sqrt{1 - (x+1)^2}. \] Simplifying the right-hand side: \[ \sqrt{1 - (x^2 + 2x + 1)} = \sqrt{2x - x^2}. \] Both sides match, verifying \( h = 1 \).

Key Concepts

Understanding Function Transformation

In mathematics, transforming a function involves modifying its behavior or shifting its position on a graph. This is commonly done by changing its formula. Function transformation is essential because it shows us how different variables influence the graph of the function.

Some common transformations include:

• Vertical shifts: This occurs when a constant is added or subtracted to the entire function, moving the graph up or down.
• Horizontal shifts: This involves adding or subtracting a constant directly to the variable, moving the graph left or right.
• Reflections: By multiplying by a negative, you can reflect the graph over the x-axis or y-axis.
• Stretching and Compressing: This occurs when you scale the function by multiplying it by a constant, altering its steepness.

These transformations help us model real-world scenarios by adjusting the graph to fit new conditions or assumptions.

Mastering Constant Horizontal Shifts

A constant horizontal shift in a function signifies moving the graph of the function left or right without changing its shape. Consider a function, say \(f(x)\), and a modified version \(f(x + h)\). The \(+h\) or \(-h\) represents a horizontal translation.

Here's a simple breakdown of how this works:

• Rightward Shift: If \(h > 0\), the graph of \(f(x)\) slides to the right by \(h\) units.
• Leftward Shift: If \(h < 0\), the graph moves to the left by \(|h|\) units.

Such shifts do not change any of the function's characteristics like steepness, symmetry, or maximum/minimum points; they merely reposition it along the x-axis.

This concept comes into play when comparing our two functions \(f(x) = \sqrt{1 - x^2}\) and \(g(x) = \sqrt{2x - x^2}\). Solving \(g(x) = f(x+h)\) by finding \(h = 1\) implies a one-unit shift to the right.

Solving Equations with Square Roots

Equations involving square roots can look intimidating at first, but they can be tackled with systematic methods. The process usually involves isolating the square root term and then squaring both sides to eliminate the root.

Here's a step-by-step guide to solving such equations:

• Isolate the Square Root: Make sure that one side of the equation is entirely the square root term.
• Eliminate the Square Root: Square both sides of the equation to remove the square root. Be mindful of both solutions, as squaring can introduce extraneous solutions.
• Simplify and Solve: Rearrange the resultant equation to solve for the unknown variable, like \(h\) in our example.
• Verify Solutions: Substitute back into the original equation to ensure the roots are valid."

In our exercise, simplifying the equation and squaring both sides led us to identify that \(h = 1\), representing a perfect one-unit horizontal shift that achieves the transformation from \(f(x)\) to \(g(x)\).