Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{2} x, \quad g(x)=4-\log _{2} x$$

Step-by-Step Solution

Answer

The graph of \(g(x)=4-\log _{2} x\) is the reflection of the graph of \(f(x)=\log_{2} x\) about the x-axis followed by a shift upwards by 4 units.

Step 1: Identify the Base Function

The base function or the starting point is \(f(x)=\log_{2} x\). It is a logarithmic function.

Step 2: Identify the Transformation Functions

We can rewrite \(g(x)\) by rearranging terms as \(g(x)=-\log_{2} x + 4\). Here, the transformation of the base function involves two operations: a reflection due to the negative sign and a vertical shift due to the addition of 4.

Step 3: Explain each Transformation

The negative sign means the graph of \(f(x)\) is reflected about the x-axis. This means flipping the graph upside down. The constant 4, being added to the function, shows a vertical shift upwards by 4 units. Thus, the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) by flipping the graph of \(f\) about the x-axis and then shifting it upward by 4 units.

Key Concepts

Logarithmic Functions

Logarithmic functions, like the one in our base function \(f(x)=\log_{2}{x}\), are the inverse of exponential functions. They are used to determine the exponent needed to obtain a certain value when the base is known. For example, in the context of base 2, \(\log_{2}{8}=3\) because \(2^3=8\). The graph of a basic logarithmic function \(y=\log_{b}{x}\) is characterized by a vertical asymptote along the y-axis and a curve that increases slowly to the right, never touching the asymptote.

The logarithmic function graph is not symmetrical and only exists for positive values of \(x\), because you cannot take the logarithm of a negative number or zero. The importance of understanding logarithmic functions lies in their broad application across various fields such as science, engineering, and finance where they model phenomena such as acoustic intensity and compound interest.

Reflection of Graphs

Reflection of graphs is a transformation that flips a graph over a specific axis. In the case of the function \(g(x)=4-\log_{2}{x}\), the \(\log_{2}{x}\) part is negated, denoted as \( - \log_{2}{x} \). This negation reflects the graph of the base function \(\log_{2}{x}\) over the x-axis. Imagine a mirror placed along the x-axis; the reflected graph will be an exact, upside-down image of the original. Such transformations are fundamental in understanding how algebraic manipulation affects the shape and position of a graph. Reflections can occur over the y-axis as well, by negating the \(x\) variable inside the function.

Vertical Shift

A vertical shift is a translation of a graph up or down in a straight line, without changing its shape or orientation. This occurs when a constant is added to or subtracted from a function's output. For the function \(g(x)\), the addition of \(+4\) after negating \(\log_{2}{x}\) shifts the graph of the function up by 4 units. If the constant were negative, the graph would be shifted downward. This is an easy way to adjust a graph's position on the coordinate plane to suit a particular context or dataset. Vertical shifts are regularly used in real-life applications such as tracking temperature changes over time, where a baseline temperature is adjusted up or down.

Algebraic Manipulation

Algebraic manipulation involves the rearranging and simplifying of equations to solve problems or to understand function transformations better. In our step-by-step solution, algebraic manipulation was used to rearrange \(g(x)=4-\log_{2}{x}\) into \(g(x)=-\log_{2}{x} + 4\). This form made it clearer that there are two transformations applied to the base function: a reflection and a vertical shift. Learning to manipulate algebraic expressions with confidence allows students to handle more complex equations and to understand the relationship between the algebraic representation of a function and its graphical representation. This is especially useful when applying transformations to a graph, as it can help predict the effects of changes in the function's equation on its graph.