Problem 72 - Algebra for College Students

Question

Graph \(f(x)=\log _{5} x\). Now predict the graphs for \(f(x)=\) \(2 \log _{5} x, f(x)=-4 \log _{5} x\), and \(f(x)=\log _{5}(x+4)\). Graph these three functions on the same set of axes with \(f(x)=\log _{5} x\).

Step-by-Step Solution

Answer

The graphs of the four functions show different transformations: basic, vertical stretch, vertical stretch and reflection, and horizontal shift.

Step 1: Understanding the Basic Logarithmic Graph

The function \(f(x) = \log_5 x\) is a logarithmic function with base 5. The basic shape of this graph is an increasing curve that passes through the point (1,0) because \(\log_5(1) = 0\) and approaches zero as x approaches infinity. It is undefined for \(x \leq 0\).

Step 2: Graphing \(f(x) = \log_5 x\)

Plot the points (1,0), (5,1), and (25,2), where each value of y corresponds to \(5^y=x\). Draw a smooth curve that passes through these points and approaches the vertical axis but never touches it.

Step 3: Transform \(f(x) = 2 \log_5 x\)

The function \(f(x) = 2 \log_5 x\) is a vertical stretch of the original log function by a factor of 2. This means each value of \(f(x)\) is doubled. For example, if \(\log_5(5) = 1\), then \(2\log_5(5) = 2\). Plot points like (1,0), (5,2), (25,4) and draw a curve.

Step 4: Transform \(f(x) = -4 \log_5 x\)

The function \(f(x) = -4 \log_5 x\) involves a vertical stretch and a reflection over the x-axis. Multiply each value of \(f(x)\) by -4. If \(\log_5(5) = 1\), then \(-4\log_5(5) = -4\). Plot points such as (1,0), (5,-4), (25,-8) and draw the curve.

Step 5: Transform \(f(x) = \log_5(x+4)\)

This transformation is a horizontal shift to the left by 4 units. Therefore, the graph \(\log_5(x+4)\) is the same as \(\log_5 x\) but shifted left. Key points are adjusted; instead of (1,0), use (-3,0), adjust other points like (1,1), and plot the shifted curve.

Step 6: Graphing all the functions together

On the same set of axes, plot all four curves: the original \(f(x) = \log_5 x\), the stretched \(f(x) = 2\log_5 x\), reflected and stretched \(f(x) = -4\log_5 x\), and the shifted \(f(x) = \log_5(x+4)\). Make sure each curve correctly reflects the transformation described in each title step.

Key Concepts

Logarithmic Transformations

Logarithmic transformations involve changing the basic properties of a logarithmic function, such as shifting, stretching, or reflecting it. Understanding these transformations helps you graph various forms of logarithmic functions efficiently.

• Basic Logarithmic Plotting: The function \( f(x) = \log_5 x \) serves as a baseline. It passes through crucial points like (1,0), (5,1), and (25,2), and is always increasing for positive x.
• Transformation Types: These include vertical stretches, reflections, and horizontal shifts. Each modifies the graph's appearance while maintaining certain key characteristics.
• Analyzing New Functions: For example, knowing that \( f(x) = 2 \log_5 x \) stretches each value by a factor of 2 allows easier visualization on shared axes.

By connecting these transformations back to the original function, plotting becomes more intuitive.

Vertical Stretch

A vertical stretch involves multiplying the function's output by a constant, effectively stretching the graph along the y-axis. When graphing \( f(x) = 2 \log_5 x \), for instance, every y-value from the base function \( f(x) = \log_5 x \) is doubled. This results in a steeper curve.

• Effect of Stretching: The graph remains undefined at x ≤ 0, still passing through the point (1,0). However, points like (5,1) from the original graph become (5,2) because of the stretch.
• Identifying Stretched Points: Calculate stretched points by multiplying the original y-values by the stretch factor. If you originally had \( y = 1 \), it changes to \( y = 2 \) after a stretch by a factor of 2.

Keep the x-intercepts untouched when stretching vertically, which helps in maintaining overall graph integrity.

Horizontal Shift

A horizontal shift moves the logarithmic graph left or right, impacting the values where the function is defined. Consider \( f(x) = \log_5(x + 4) \), a function shifted 4 units to the left compared to the basic \( \log_5 x \).

• Understanding Shift Direction: The expression inside the log, \( x+4 \), indicates moving left by 4 units. For \( f(x) = \log_5(x - 4) \), it would shift right.
• New Key Points: Points like (1,0) in \( \log_5 x \) shift to (-3,0) in \( \log_5(x+4) \). Similarly, (5,1) becomes (1,1).

These adjustments form the basis of your new graph outline, retaining similar shape but starting from a different part of the x-axis.

Reflection of Graphs

Graph reflections involve flipping a graph over a central axis. With \( f(x) = -4 \log_5 x \), the reflection occurs over the x-axis, along with a vertical stretch by a factor of 4.

• Reflection and Stretch Combined: Each originally positive y-value of \( \log_5 x \) becomes negative and quadrupled. As an example, where \( \log_5(5) = 1 \), the transformed graph intersects at (5,-4).
• Impact of the Reflection: The entire graph orientation changes. This flips the graph upside down and exaggerates point distance from the x-axis.

Hence, reflection offers a new perspective on familiar graphs, reversing their increases or decreases while stretching them proportionally.