Question

Describe how the graph of each logarithmic function could be obtained from the graph of \(y=\log _{3} x\). a) \(y=5 \log _{3}(-4 x+12)-2\) b) \(y=-\frac{1}{4} \log _{3}(6-x)+1\)

Short answer

For (a), apply horizontal reflection, shift, and compression, stretch vertically, and shift down. For (b), apply horizontal reflection with shift, compress vertically, reflect over x-axis, and shift up.

Step-by-step solution

Identify the Base Graph

The base graph is given as \(y = \log_{3}(x)\). This will be used as a reference to derive the other graphs.

Analyze Function (a)

For function (a) \(y = 5 \log_{3}(-4x + 12) - 2\):\1. Identify the transformations inside the logarithm: \(-4x + 12\). This implies a horizontal shift and a reflection over the y-axis.\2. Factor the inner term: \(-4(x - 3)\). This suggests a horizontal shift 3 units to the right and a reflection over the y-axis due to the negative sign.\3. The multiplication by 4 indicates a horizontal compression by a factor of \frac{1}{4}\.\4. Multiply the logarithm by 5 results in a vertical stretch by a factor of 5.\5. Subtract 2 for a downward vertical shift by 2 units.

Derive the Graph for (a)

Starting from \(y = \log_{3}(x)\):\1. Reflect it horizontally: \(y = \log_{3}(-x)\).\2. Apply horizontal shift: \(y = \log_{3}(-4(x - 3))\).\3. Apply horizontal compression: \(y = \log_{3}(-4x + 12)\).\4. Vertical stretch: \(y = 5 \log_{3}(-4x + 12)\).\5. Final vertical shift: \(y = 5 \log_{3}(-4x + 12) - 2\).

Analyze Function (b)

For function (b) \(y = -\frac{1}{4} \log_{3}(6 - x) + 1\):\1. Identify the transformations inside the logarithm: \(6 - x\), which implies a reflection over the y-axis and a horizontal shift.\2. Factor the inner term: \(-1(x - 6)\), which suggests a horizontal shift 6 units to the right and a reflection over the y-axis.\3. Multiply the logarithm by \frac{1}{4}\ indicates a vertical compression by a factor of 4.\4. Negate the logarithm to reflect it over the x-axis.\5. Add 1 for a vertical shift upward by 1 unit.

Derive the Graph for (b)

Starting from \(y = \log_{3}(x)\):\1. Horizontal shift and reflection: \(y = \log_{3}(6 - x)\).\2. Vertical compression: \(y = \frac{1}{4} \log_{3}(6 - x)\).\3. Reflection over the x-axis: \(y = -\frac{1}{4} \log_{3}(6 - x)\).\4. Final vertical shift: \(y = -\frac{1}{4} \log_{3}(6 - x) + 1\).

Key concepts

Logarithmic Function Graphing

To graph a logarithmic function, start with the basic graph of a parent function, which is usually represented as \(y = \log_b(x)\). This graph has the following properties:
- It passes through the point (1,0) because log base any number of 1 is 0.
- It approaches the y-axis (vertical asymptote) but never touches it.
The y-axis serves as a vertical asymptote.
- It increases slowly to infinity as x increases.
Understanding the basic shape of this parent function is critical, as all other transformations will be compared against this.

Horizontal Shift

A horizontal shift occurs when a constant, \(c\), is added or subtracted inside the logarithmic function. The general form looks like \(y = \log_b(x - c)\) for a shift to the right and \(y = \log_b(x + c)\) for a shift to the left.
For example, in the function \(y = \log_3(x - 3)\), the graph shifts 3 units to the right. Always remember:
- If \(c\) is positive, the graph shifts to the right.
- If \(c\) is negative, the graph shifts to the left.

Vertical Shift

A vertical shift happens when a constant, \(d\), is added or subtracted from the entire logarithmic function. The form is \(y = \log_b(x) + d\) for an upward shift and \(y = \log_b(x) - d\) for a downward shift.
For instance, in the function \(y = \log_3(x) - 2\), the graph shifts 2 units down. Key points:
- Upward shift if you're adding \(d\).
- Downward shift if you're subtracting \(d\).

Reflection Over Axes

Reflections flip the graph over a given axis. There are two types to consider:
1. Reflection over the y-axis: This occurs when you replace \(x\) with \(-x\). For example, \(y = \log_3(-x)\) is a reflection over the y-axis.
2. Reflection over the x-axis: This involves multiplying the entire logarithmic function by -1. For example, \(y = -\log_3(x)\) reflects the graph over the x-axis.

Vertical Stretch and Compression

A vertical stretch or compression changes the steepness of the graph. Multiplying the logarithmic function by a factor, \(k\), modifies it vertically. For example:
- \(y = k \log_b(x)\) stretches the graph vertically if \(k > 1\) and compresses it if \(0 < k < 1\).
For instance, in \(y = 5 \log_3(x)\), the graph is stretched vertically by a factor of 5. Similarly, in \(y = \frac{1}{4} \log_3(x)\), the graph compresses vertically by a factor of \(\frac{1}{4}\).

Horizontal Stretch and Compression

Horizontal stretches or compressions involve changing the \(x\) variable inside the logarithm by a constant factor. This modification is generally of the form \(y = \log_b(kx)\) where \(k\) impacts the stretch/compression:
- \(y = \log_b(kx)\): If \(k > 1\), it compresses the graph horizontally.
- \(y = \log_b(\frac{x}{c})\): If \(0 < k < 1\), it stretches the graph horizontally.
Example: For \(y = \log_3(-4x)\), there is a horizontal compression by a factor of \(\frac{1}{4}\).