Question

Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(f(x)=\log _5 x\); reflection in the \(x\)-axis, followed by a translation 9 units left

Short answer

The rule for the function \(g\) that represents the given transformations of the function \(f(x) = \log_5x\) is \(g(x) = -\log_5(x+9)\)

Step-by-step solution

Apply Reflection

Firstly, perform the reflection in the x-axis, this is achieved by taking the negative of the function. Thus, the transformation results in \(g(x) = -\log_5x\)

Apply Translation

Secondly, perform a translation of 9 units to the left. This is done by subtracting 9 from the input of the function. Thus, the final transformation results in \(g(x) = -\log_5(x+9)\). The negative sign is for the reflection and the 'x+9' part is for the left shift.

Key concepts

Logarithmic Function Reflection

Understanding the reflection of logarithmic functions is key to mastering graph transformations in algebra. In our exercise, the reflection is done across the x-axis. To visualize this, imagine a mirror placed right on the x-axis; the graph of the logarithmic function would then appear upside down. Mathematically, reflecting a function across the x-axis can be achieved by multiplying the function by -1.

So if we start with the base function, in this case, \(f(x) = \log_5 x\), the reflection is represented by \(g(x) = -f(x)\) which gives us \(g(x) = -\log_5 x\). This transformation inverts every point on the graph; heights become depths, and vice versa, effectively flipping the graph visually while preserving the x-axis positions of the points.

Logarithmic Function Translation

Translation involves moving a graph horizontally or vertically, without altering its shape or orientation. When translating a logarithmic function, it's important to consider the direction and magnitude of the shift. In the given exercise, we're asked to translate the graph 9 units left after the reflection.

To translate the function left by 9 units, we adjust the input variable in the function. Translating a function \(f(x)\) by \(k\) units to the left involves replacing \(x\) with \(x+k\). So for our function, \(g(x) = -\log_5(x+9)\). It's crucial to note that \(x+9\) is within the logarithmic function's argument, showing that the entire function moves 9 units left, not just the x-variable.

Logarithmic Function Transformation Rules

To master function transformation, especially with logarithms, there are a set of rules to remember. Reflecting a function flips it over a specified axis, translating a function shifts it in a specific direction, and there are rules for stretching and compressing as well.

Here are the main transformation rules applied to a base logarithmic function \(f(x) = \log_b(x)\):

• Reflection across the x-axis: \(g(x) = -\log_b(x)\)
• Reflection across the y-axis: \(g(x) = \log_b(-x)\)
• Horizontal shift k units right: \(g(x) = \log_b(x - k)\)
• Horizontal shift k units left: \(g(x) = \log_b(x + k)\)
• Vertical shift k units up: \(g(x) = \log_b(x) + k\)
• Vertical shift k units down: \(g(x) = \log_b(x) - k\)

Applying these rules consistently will allow students to tackle any transformation of logarithmic functions with confidence.

Graph Transformations in Algebra 2

Graph transformations in Algebra 2 involve translating, reflecting, stretching, and compressing the graphs of functions. Understanding these transformations allows students to dissect complex functions into a series of simpler steps. When you transform the graph of a function, you're changing its position or shape without affecting the relationship between the variables.

These transformations are often taught through the lens of 'parent functions', such as linear, quadratic, exponential, and logarithmic functions. Using transformation rules, we can easily predict and sketch the outcome of applying a series of transformations to these parent graphs. The key to success with graph transformations is practice and visualizing how each rule affects the graph, so don't shy away from sketching out examples to solidify your understanding.