Problem 14 - Algebra and Trigonometry

Question

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=f(2 x)-1 \quad\) (b) \(y=2 f\left(\frac{1}{2} x\right)\)

Step-by-Step Solution

Answer

The graph of \(y=f(2x)-1\) is horizontally compressed by 2 and shifted down 1 unit. The graph of \(y=2f(\frac{1}{2}x)\) is horizontally stretched by 2 and vertically stretched by 2.

Step 1: Understand the effect of horizontal scaling

The expression \(f(2x)\) involves a horizontal scaling. When you replace \(x\) by \(2x\) in the function \(f\), it compresses the graph horizontally. This means the graph is squeezed by a factor of 2 along the x-axis. Each point \((x, y)\) on the original graph moves to \((x/2, y)\) on the new graph.

Step 2: Apply vertical shift

The term \(-1\) in \(y=f(2x)-1\) indicates a vertical shift. Subtracting 1 shifts the entire graph downward by 1 unit. Thus, each point on the graph of \(f(2x)\) moves to \((x/2, y-1)\).

Step 3: Understand the effect of horizontal stretching

In the function \(f\left(\frac{1}{2}x\right)\), replacing \(x\) by \(\frac{1}{2}x\) results in a horizontal stretch. This stretches the graph horizontally by a factor of 2. Each point \((x, y)\) on the original graph moves to \((2x, y)\).

Step 4: Apply vertical scaling

The expression \(2f\left(\frac{1}{2}x\right)\) involves a vertical scaling by a factor of 2. Multiplying by 2 stretches the graph vertically. Therefore, each point moves to \((2x, 2y)\). This means every y-coordinate is doubled.

Key Concepts

Horizontal Scaling

Horizontal scaling modifies the width of a graph along the x-axis. When you see a function like \( f(2x) \), it compresses the graph horizontally. This occurs because each x-value in the original function is multiplied by 2, making the graph appear squeezed. So every point \((x,y)\) from the original graph becomes \((x/2,y)\). In simpler terms:

• The graph shrinks toward the y-axis
• The x-coordinates of points are halved
• Overall graph width is reduced by a factor of 2

Remember, horizontal scaling is about changing the spread along the x-axis, not the height.

Vertical Shifts

Vertical shifts are straightforward. They move the entire graph up or down without altering its shape. When you subtract a constant from a function, such as in \( f(x) - 1 \), it shifts the graph downward. For example, each point \((x,y)\) of the original graph would translate to \((x, y-1)\). Here's a breakdown:

• Graph moves up when a positive number is added
• Graph moves down when a negative number is subtracted
• The shape and width of the graph remain constant

A vertical shift prevents any distortion in the graph—it just relocates it vertically.

Horizontal Stretching

Horizontal stretching expands the graph along the x-axis. With \( f\left(\frac{1}{2}x\right) \), the graph stretches horizontally. Why? Because the x-values are effectively made larger by dividing each \(x\) by 0.5 (or multiplying by 2), turning \((x, y)\) into \((2x, y)\). Key points:

• The graph expands away from the y-axis
• Each x-coordinate is now double its original
• This operation does not affect the y-values

In essence, horizontal stretching enlarges the distance between points along the x-axis while keeping the graph's height intact.

Vertical Scaling

Vertical scaling affects how tall or short your graph appears. For instance, if we multiply a function by 2, as in \( 2f(x) \), we scale it vertically. This translates every point \((x, y)\) into \((x, 2y)\), effectively doubling the y-values.Important aspects:

• Upward or downward stretching based on the scale value
• The y-coordinates multiply by the scale factor
• Broadening or narrowing the graph without horizontal impact

Vertical scaling adjusts the graph's height without altering its position along the x-axis, making it taller or shorter depending on the scaling factor.