Problem 12 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=3 f(2 x)-1 $$

Step-by-Step Solution

Answer

The point (1, -10) is on the graph of the transformed function.

Step 1: Understanding the Transformation Function

We start by identifying the transformation applied to the function. The given function is transformed to \( y = 3f(2x) - 1 \). This function involves vertical scaling by a factor of 3, horizontal scaling due to the factor of 2 inside \( f(x) \), and a vertical shift by -1.

Step 2: Identify the Original Function Point

The original point given is \((2, -3)\) on the graph of \( y = f(x) \). This means that when \( x = 2 \), \( f(x) = -3 \), so \( f(2) = -3 \).

Step 3: Horizontal Compression

The horizontal transformation inside \( f(2x) \) compresses the graph horizontally by a factor of \( \frac{1}{2} \). Therefore, the x-coordinate must be revised by multiplying by \( \frac{1}{2} \). Thus, the new x-coordinate becomes \( x = 1 \).

Step 4: Apply Function Transformation

Substitute the x-value after horizontal compression (\( x = 1 \)) into the function \( f(2x) \). Since \( f(2) = -3 \), \( f(2 \times 1) = -3 \). Now calculate \( y = 3(-3) \) according to the vertical scaling factor of 3. We obtain \( y = -9 \).

Step 5: Vertical Shift Adjustment

Apply the final vertical shift by subtracting 1 to the \( y \) value from the previous step. Thus, \( y = -9 - 1 = -10 \).

Step 6: Determine the New Transformed Point

Combining the new x-coordinate from the horizontal compression and the adjusted y-coordinate, the new transformed point on the graph of \( y = 3f(2x) - 1 \) is \((1, -10)\).

Key Concepts

Vertical Scaling

Vertical scaling of a function occurs when you multiply the entire function by some constant. For the equation \( y = 3f(x) \), the function \( f(x) \) is vertically scaled by a factor of 3. This means that every y-coordinate of the function is scaled by 3 times its original size. A positive scaling factor greater than 1 will stretch the graph upward if \( f(x) \) is positive or downward if \( f(x) \) is negative.

• If the original y-value is positive, like 4, vertical scaling by 3 makes it 12.
• If the original y-value is negative, like -2, it becomes -6.

Vertical scaling does not affect the x-values at all, leaving them unchanged. This transformation changes the steepness and height of the graph but not its shape.

Horizontal Scaling

Horizontal scaling involves changing the x-values of the function by multiplying them by a constant. In our equation, \( y = 3f(2x) \), the factor within the function \( 2x \) indicates horizontal transformation. Here, x-values are affected by a factor of 2. However, the factor directly applied in the transform is \( \frac{1}{2} \), which implies compression.

• Initially, if x = 2, you adjust it to \( x \times \frac{1}{2} = 1 \).
• Imagine shrinking the graph horizontally by the factor used within the function.

This compression effectively shifts all x-values closer to the y-axis, compressing the graph's width. Understanding this helps in visualizing how distances along the x-axis are modified.

Vertical Shift

Vertical shift involves moving the entire graph up or down. This is achieved by adding or subtracting a constant to the function. In the expression \( y = 3f(2x) - 1 \), the \(-1\) at the end denotes a vertical shift of one unit downwards.

• If you have a point (x, y), with a vertical shift downward by 1, the new y-value is \( y - 1 \).
• This mirrors shifting your entire function lower on the y-axis.

Vertical shifts do not affect the x-values at all, only the y-values. It's comparable to shifting a landscape up or down without stretching it in any direction.

Horizontal Compression

Horizontal compression occurs when the x-coordinates are scaled by a value larger than 1. In the function \( y = 3f(2x) \), the factor inside \( f(x) \) is \( 2 \), meaning the x-coordinates are compressed by \( \frac{1}{2} \).

• This takes the point (2, -3) and compresses it to (1, -3) horizontally.
• Compressing horizontally means moving points closer to each other along the x-axis.

Horizontal compression creates a tighter cluster of x-values, effectively squeezing the graph's width without altering its vertical attributes. Understanding horizontal compression is critical in predicting how the appearance of the function changes along the x-axis.