Problem 43 - Algebra and Trigonometry

Question

Sketch the graph of the function by plotting points. $$ f(x)=2 \log x $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 2 \log x \) passes through (1,0), (10,2), (0.1,-2), and (0.01,-4), showing a vertical stretch of a logarithmic curve.

Step 1: Understand the Function

The function given is \( f(x) = 2 \log x \). This function is a logarithmic function with a vertical stretch by a factor of 2. The basic logarithmic function is \( \log x \), which passes through the point (1,0) and increases slowly as \( x \) increases. The factor of 2 will stretch this function upwards.

Step 2: Choose Points to Plot

Select a few values of \( x \) to plug into the function \( f(x) = 2 \log x \). For example, choose \( x = 1, 10, 0.1, \) and \( 0.01 \). These are values where the logarithm has simple results, which will help us plot the graph efficiently.

Step 3: Calculate \( f(x) \) Values

Compute \( f(x) \) for the chosen values:- For \( x = 1 \), \( f(1) = 2 \log 1 = 2 \times 0 = 0 \)- For \( x = 10 \), \( f(10) = 2 \log 10 = 2 \times 1 = 2 \)- For \( x = 0.1 \), \( f(0.1) = 2 \log 0.1 = 2 \times (-1) = -2 \)- For \( x = 0.01 \), \( f(0.01) = 2 \log 0.01 = 2 \times (-2) = -4 \)

Step 4: Plot the Points

On a set of axes, plot the points \((1, 0)\), \((10, 2)\), \((0.1, -2)\), and \((0.01, -4)\). The point (1, 0) is on the x-axis, (10, 2) is higher up, (0.1, -2) and (0.01, -4) are below the x-axis, showing how the graph approaches negative infinity as \( x \) approaches zero.

Step 5: Sketch the Graph

Draw a smooth curve through the plotted points which displays the nature of a logarithmic graph. As \( x \) increases, the graph should rise, and as \( x \) decreases towards zero, the graph should drop sharply downwards to negative infinity, showing the steep decrease of the logarithm near zero.

Key Concepts

Graph Sketching

When sketching the graph of a logarithmic function, it's important to understand the overall shape and behavior of the graph. The basic form of a logarithmic graph, like \( \log x \), is a curve that increases slowly as \( x \) increases and passes through the point (1,0). For the function \( f(x) = 2 \log x \), the overall shape remains similar, but with vertical stretching due to the multiplier of 2.
This means for each point on the graph of \( \log x \), the corresponding \( y \)-value is multiplied by 2 in the graph of \( f(x) = 2 \log x \).
This results in the curve rising more steeply as \( x \) gets larger. As with all logarithmic graphs, the graph will never touch the \( x \)-axis and will fall steeply as \( x \) approaches 0.

• The vertical stretch makes the curve increase more pronounced as \( x \) increases.
• The graph will pass through the point (1,0) as \( \log 1 = 0 \).
• It will slope quickly downward as it approaches the vertical asymptote at \( x = 0 \).

Function Transformation

Function transformation includes any modification made to the basic graph of a function. In the case of \( f(x) = 2 \log x \), it's important to identify the type of transformation applied. Here, the transformation is a vertical stretch by a factor of 2.
This does not affect the horizontal behavior of the graph or its vertical asymptote, which remains at \( x = 0 \). However, it does change the way the graph rises and falls.

• Vertical stretching alters the steepness without shifting the graph left or right.
• All points on the graph are moved further from the \( x \)-axis by a factor of 2.
• The graph still retains the basic properties of a \( \log \) function, such as the basic point (1,0).

Understanding these transformations helps in anticipating the changes in graph appearance and behavior, aiding in accurate sketching.

Plotting Points

Plotting points is a fundamental step in graphing a function manually. By choosing strategic values of \( x \) that provide simple results, you can more easily sketch the curve of the function. Start by selecting a mix of positive values: the basic point (1,0), along with values like \( x = 10, 0.1, \) and \( 0.01 \).
Let's see how these work for \( f(x) = 2 \log x \).

• At \( x = 1 \), \( f(1) = 2 \times \log 1 = 0 \), which shows the point (1,0).
• For \( x = 10 \), \( f(10) = 2 \times \log 10 = 2 \), which defines the point (10,2).
• When \( x = 0.1 \), \( f(0.1) = -2 \), leading to the point (0.1,-2).
• And at \( x = 0.01 \), \( f(0.01) = -4 \), leading to the point (0.01,-4).

Once all points are calculated and plotted, it becomes much easier to draw a smooth curve through them, completing the sketch.