Problem 74 - Precalculus Mathematics for Calculus

Question

Shifting, Shrinking, and Stretching Graphs of Functions Let \(f(x)=x^{2} .\) Show that \(f(2 x)=4 f(x),\) and explain how this shows that shrinking the graph of \(f\) horizontally has the same effect as stretching it vertically. Then use the identities \(e^{2+x}=e^{2} e^{x}\) and \(\ln (2 x)=\ln 2+\ln x\) to show that for \(g(x)=e^{x}\) a horizontal shift is the same as a vertical stretch and for \(h(x)=\ln x\) a horizontal shrinking is the same as a vertical shift.

Step-by-Step Solution

Answer

Compressing \(f(x)\) horizontally by a factor of 2 results in a vertical stretch by 4. For \(g(x)\), horizontal shift equals vertical stretch; for \(h(x)\), horizontal shrink equals vertical shift.

Step 1: Analyze the function transformation

For the function \(f(x) = x^2\), we need to verify the transformation relationship. Substitute \(2x\) into \(f\) to get \(f(2x)\). This results in \((2x)^2 = 4x^2 = 4f(x)\). Thus, \(f(2x) = 4f(x)\), which shows that compressing the function horizontally by a factor of 2 results in stretching it vertically by a factor of 4.

Step 2: Examine horizontal shift for exponential function

Consider \(g(x) = e^x\). We know that \(e^{2+x} = e^2 \, e^x\). Here, shifting \(g(x)\) horizontally by 2 (\(g(x+2)\)) is same as multiplying \(g(x)\) by \(e^2\), which represents a vertical stretch by a factor of \(e^2\).

Step 3: Analyze horizontal shrinking for logarithmic function

For \(h(x) = \ln x\), we know \(\ln (2x) = \ln 2 + \ln x\). A horizontal compression by a factor of 2 (expressed as \(h(2x)\)) leads to an addition of \(\ln 2\) to \(h(x)\), which represents a vertical shift upwards by \(\ln 2\).

Key Concepts

Horizontal Shifts

When you think of horizontal shifts in graph transformations, envision moving the entire graph left or right along the x-axis. Imagine you have a function, let's say \( g(x) = e^x \). If we alter it to \( g(x+2) \), the graph shifts 2 units to the left. This transformation effectively results in the same output by changing the magnitude of the function vertically.
Horizontal shifts can often be misunderstood at first, but they play a crucial role in understanding function transformations:

• Shifting left involves adding to the input \( x \), as in \( f(x+c) \).
• Shifting right involves subtracting from the input \( x \), as in \( f(x-c) \).

Understanding horizontal shifts will greatly improve your grasp on the effects of moving functions graphically and what that means for transformations overall.

Vertical Stretches

Vertical stretches adjust the height of a graph. They expand or compress the graph along the y-axis. For the quadratic function \( f(x) = x^2 \), taking \( f(2x) \) and showing that it's equal to \( 4f(x) \) illustrates a vertical stretch. Essentially, horizontally shrinking the x-values by a factor of 2 causes a vertical stretch factor of 4.
Here's how you can visualize vertical stretches:

• Multiplying a function by a constant \( k \) greater than 1 (like \( 4, 5 \), etc.) stretches the graph vertically.
• If \( 0 < k < 1 \), the graph compresses vertically.

Vertical stretches are vital when considering how functions change in amplitude or visibility in practical applications.

Function Shrinking

Shrinking functions, especially horizontally, can drastically alter their appearance by compressing them along the x-axis. This concept complements our understanding of vertical stretches. If you compress the function \( f(x) = x^2 \) by replacing \( x \) with \( 2x \), the output transforms from \( f(x) \) to \( f(2x) = 4f(x) \). This technique indicates doubling the input halves the function’s x-values.
Function shrinking offers insights into:

• How input manipulation affects overall graph width.
• The invisible yet potent scaling effects on function curves.

Grasping function shrinking leads to proficient graph manipulation and can reveal deeper insights into solving equation-based problems graphically.

Logarithmic Functions

Logarithmic functions showcase a unique perspective on transformations, particularly with compressions and shifts. For instance, with \( h(x) = \ln x \), we can analyze its transformation: \( \ln (2x) = \ln 2 + \ln x \), demonstrating that a horizontal shrinking results in a vertical shift upwards by \( \ln 2 \).
Working with logarithmic functions entails:

• The addition property: \( \ln(ab) = \ln a + \ln b \), crucial for breaking down complex logarithmic expressions.
• The effects of translating horizontally and translating vertically.

Understanding logarithmic transformations strengthens your capability to solve logarithmic equations and graph intricate logarithmic functions effortlessly.