Problem 47 - A Graphical Approach to College Algebra

Question

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=3 \log _{2} x+1$$

Step-by-Step Solution

Answer

Stretch vertically by 3 and shift up by 1 unit.

Step 1: Understand the Base Function

The base function is \( y = \log_{2} x \), which is a logarithmic function with base 2. It's important to understand that this graph passes through the point \((1,0)\), and increases very slowly as \(x\) increases.

Step 2: Apply Vertical Stretch

The given function is \( y = 3 \log_{2} x + 1 \). The coefficient 3 in front of the logarithmic term indicates a vertical stretch of the graph of \( y = \log_{2} x \) by a factor of 3. This means all values of \( y \) on the original graph are multiplied by 3.

Step 3: Apply Vertical Translation

The "+1" in the function \( y = 3 \log_{2} x + 1 \) represents a vertical translation. It shifts the entire stretched graph of \( y = 3 \log_{2} x \) upwards by 1 unit.

Step 4: Graph the Transformed Function

To graph \( y = 3 \log_{2} x + 1 \), begin by plotting points from the base function and apply the transformations. Stretch these points vertically by a factor of 3, then move each point up by 1 unit. The original asymptote along the y-axis remains unchanged.

Key Concepts

Graph Transformations

Understanding how functions change visually involves graph transformations. When we transform graphs, we're adjusting their shape or position without altering their core identity. For logarithmic functions, these transformations can include shifts, stretches, and even reflections.

In this exercise, the function we're dealing with is essentially a transformed version of the base logarithmic function \( y = \log_{2} x \). Initially, this base graph passes through the point \((1, 0)\) and has the y-axis as a vertical asymptote. When transformations are applied, we start by imagining these changes happening step-by-step.

• First, identify any multipliers that might stretch or compress the graph vertically or horizontally.
• Look for additional constants that might shift the graph up, down, left, or right.

Breaking transformations into these clear steps helps in visualizing how the base graph morphs into the final image presented by the function \( y = 3 \log_{2} x + 1 \).

Vertical Stretch

A vertical stretch changes how steep a graph appears. This transformation involves multiplying all \( y \)-values by a constant factor. Imagine pulling the graph upward or pushing it downward, making it taller or shorter.

For the function \( y = 3 \log_{2} x \), the multiplier is 3. This means:

• Each point on the base graph is moved three times further from the x-axis.
• This stretching effect amplifies how rapidly the function increases, though the x-coordinates remain unchanged.

Visualizing this, the graph appears much steeper than the original \( y = \log_{2} x \) graph. By simply multiplying, we've altered the graph's growth scale dramatically.

Vertical Translation

Vertical translation is about shifting the entire graph up or down. It's like moving a drawing on a piece of paper without rotating or resizing it.

In \( y = 3 \log_{2} x + 1 \), the '+1' shifts every point on the graph of \( y = 3 \log_{2} x \) upwards by 1 unit.

• This changes the vertical position of all plotted points.
• The asymptote of the graph, aligning along the y-axis, remains consistent, as the x-values are not affected.

After the stretch, this translation finalizes the graph's position, giving us a complete transformation of the original function. The end graph is now a vertically stretched and upward-shifted version of the initial \( \log_{2} x \) graph.