Problem 40 - Calculus Single Variable

Question

In Exercises \(40-43,\) 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 \(g\) ? \(f(x)=2 x+1, g(x)=2 x+5\)

Step-by-Step Solution

Answer

The constant \(h\) is 2, representing a horizontal left shift of 2 units.

Step 1: Understand the Problem

We need to find a constant \(h\) such that \(g(x) = f(x + h)\). This means we are looking for a translation of the function \(f\) that will result in the function \(g\). In simpler terms, we need to determine how much the graph of \(f(x)\) is shifted to obtain \(g(x)\).

Step 2: Express g(x) in terms of f

Given \(f(x) = 2x + 1\) and \(g(x) = 2x + 5\), we need \(g(x)\) to be equal to \(f(x+h)\). So, we set it up as follows: \[ f(x+h) = 2(x+h) + 1 = 2x + 2h + 1 \]

Step 3: Set up the equation

To find \(h\), set \(g(x)\) equal to \(f(x+h)\): \[ g(x) = 2x + 5 \] \[ f(x+h) = 2x + 2h + 1 \] Set these two equations equal: \[ 2x + 2h + 1 = 2x + 5 \]

Step 4: Solve for h

Since \(2x\) terms are present on both sides, they cancel out, leaving: \[ 2h + 1 = 5 \] Solve for \(h\) by subtracting 1 from both sides: \[ 2h = 4 \] Then divide by 2: \[ h = 2 \]

Step 5: Interpretation

The graph of \(f(x)\) when shifted horizontally by \(h = 2\) units to the left (since \(x+h\) means a left shift) becomes the graph of \(g(x)\).

Key Concepts

Horizontal Shift

A horizontal shift in functions changes their position left or right on the coordinate plane without altering their shape. It involves shifting the input variable, typically by a constant value. This is often expressed as adjusting the function by "adding" a constant within the function's argument: that is, instead of computing \(f(x)\), we compute \(f(x+h)\).

In mathematical terms:

• A shift to the right by \(h\) units means replacing \(x\) with \(x-h\) in the function, since \(x-h\) reaches a further place than \(x\) alone.
• A shift to the left by \(h\) units requires replacing \(x\) with \(x+h\), effectively moving the function left.

Through this horizontal shift, we align the function’s position on the graph to match our desired function transformation, maintaining its original form.

Linear Functions

Linear functions are foundational to understanding various mathematical concepts. These functions graph as straight lines, represented by the general equation \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

The slope \(m\) determines the line's steepness and direction:

• If \(m > 0\), the function rises as it moves to the right.
• If \(m < 0\), the function falls as it moves to the right.

The y-intercept \(c\) pinpoints where the function crosses the y-axis. It provides a starting value for the function when the input \(x\) is zero.

For example, with the function \(f(x) = 2x + 1\), the slope \(m\) is 2, creating a line that rises at a steady rate. The y-intercept is 1, so the graph starts at this point on the y-axis.

Function Transformation

Function transformations are systematic operations that alter the graph’s size, shape, position, or orientation. They encompass shifting, reflecting, stretching, or compressing the graph. In our example, we focused on horizontal shifts, but let's briefly overview other possibilities.

Types of transformations include:

• Vertical Shifts: Adding or subtracting a constant to the entire function \(f(x) + k\) shifts the graph up or down.


• Reflections: Multiplying by \(-1\) reflects the function across an axis (e.g., \(-f(x)\) reflects over the x-axis).


• Stretches/Compressions: Multiplying the input \(x\) or output \(f(x)\) by a constant broadens or narrows the graph.

By mastering these transformations, we gain the ability to manipulate and predict the behavior of various functions with ease, enhancing problem-solving and analytical skills.