Question

Describe, in order, a series of transformations that could be applied to the graph of \(y=\log _{7} x\) to obtain the graph of each function. a) \(y=\log _{7}(4(x+5))+6\) b) \(y=2 \log _{7}\left(-\frac{1}{3}(x-1)\right)-4\)

Short answer

a) Shift left 5, compress horizontally by 1/4, shift up 6.b) Shift right 1, reflect and stretch horizontally by 3, stretch vertically by 2, shift down 4.

Step-by-step solution

Understanding the Base Function

The base function is given as \(y = \log_{7} x\). This is the logarithmic function with base 7.

Part (a) Transformations

To transform \(y = \log_{7} x\) into \(y = \log_{7}(4(x+5)) + 6\), follow these steps:1. **Horizontal Shift:** Shift the graph to the left by 5 units. This transformation comes from the term \(x+5\) inside the logarithm.2. **Horizontal Compression:** Compress the graph horizontally by a factor of \(\frac{1}{4}\). This transformation comes from the coefficient 4 inside the logarithm.3. **Vertical Shift:** Shift the graph upward by 6 units. This transformation comes from the constant +6 outside the logarithm.

Part (b) Transformations

To transform \(y = \log_{7} x\) into \(y = 2 \log_{7} \left( -\frac{1}{3}(x-1) \right) - 4\), follow these steps:1. **Horizontal Shift:** Shift the graph to the right by 1 unit. This transformation comes from the term \(x-1\) inside the logarithm.2. **Horizontal Stretch and Reflection:** Reflect the graph over the y-axis and then stretch horizontally by a factor of 3. This transformation arises from the factor \(-\frac{1}{3}\) inside the logarithm.3. **Vertical Stretch:** Stretch the graph vertically by a factor of 2. This transformation comes from the coefficient 2 multiplying the logarithm.4. **Vertical Shift:** Shift the graph downward by 4 units. This transformation comes from the constant -4 outside the logarithm.

Key concepts

Horizontal Shift

A horizontal shift moves the graph of a function left or right. For a logarithmic function like \(y = \log_{7} x\), if we add or subtract a constant inside the logarithm's argument, we achieve this shift.
For example, in the function \(y = \log_{7} (x + 5)\), the term \(x + 5\) indicates a horizontal shift.
Additions inside the argument shift the graph left, while subtractions shift it right. Here's a simple way to remember:

• \textbf{Left Shift:} \(y = \log_{7} (x + c)\) shifts the graph left by \(c\) units.
• \textbf{Right Shift:} \(y = \log_{7} (x - c)\) shifts the graph right by \(c\) units.

Vertical Shift

A vertical shift moves the graph up or down. This is achieved by adding or subtracting a constant outside the logarithm.
For example, in \(y = \log_{7} x + 6\), the term \(+6\) moves the graph upward by 6 units. Similarly, in \(y = \log_{7} x - 4\), \(-4\) moves the graph downward by 4 units.
Remember these guidelines:

• \textbf{Upward Shift:} Adding a constant moves the graph up.
• \textbf{Downward Shift:} Subtracting a constant moves the graph down.

This technique is essential for adjusting the positioning of the graph vertically.

Horizontal Compression

Horizontal compression means squeezing the graph closer to the y-axis. When we multiply the x-term inside the logarithm by a number greater than 1, we achieve compression.
For instance, in the function \(y = \log_{7} (4x)\), the factor 4 compresses the graph horizontally by a factor of \(\frac{1}{4}\), moving points closer together.
Here's how to understand it:

• \textbf{Compressing Factor:} \(y = \log_{7} (k \cdot x)\) compresses the graph horizontally by \(\frac{1}{k}\) if \(k > 1\).

This transformation significantly alters the shape and positioning of the graph, making it narrower.

Horizontal Stretch

Horizontal stretching spreads the graph out horizontally, making it wider. This occurs when we multiply the x-term by a number between 0 and 1 inside the logarithm.
For example, in \(y = \log_{7} (\frac{1}{3} x)\), the \(\frac{1}{3}\) factor stretches the graph by a factor of 3.
Key points to remember:

• \textbf{Stretching Factor:} For \(y = \log_{7} (k \cdot x)\), if \(0 < k < 1\), then the graph is stretched horizontally by \(\frac{1}{k}\).

This helps in obtaining a wider spread of the graph.

Reflection

Reflection flips the graph over a specific axis. For logarithmic functions, reflections typically occur over the y-axis.
In the function \(y = \log_{7} (-x)\), the negative sign inside reflects the graph across the y-axis.
We also see reflections combined with other transformations. For instance, in \(y = \log_{7} (-\frac{1}{3}(x-1))\), the graph is reflected and then stretched or compressed.
To summarize:

• \textbf{Y-axis Reflection:} \(y = \log_{7} (-x)\) flips the graph over the y-axis.

Reflections change the direction of the graph, helping to understand the inversion of input values.