Problem 13 - 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. $$ y=\frac{1}{2} f(4-x) $$

Step-by-Step Solution

Answer

The point on the graph of the transformed function is \((-2, -\frac{3}{2})\).

Step 1: Understand the Transformation Function

We need to apply the transformations given by the function \( y = \frac{1}{2} f(4-x) \) to the point \((2,-3)\), which is on the graph of \( y = f(x) \). The given function indicates a horizontal reflection, a horizontal shift, and a vertical scaling.

Step 2: Apply Horizontal Reflection and Shift

The function \(4-x\) represents a horizontal reflection and a shift of the function. The term \(4-x\) can be rewritten as \(-1(x-4)\), which indicates a reflection across the vertical line and a horizontal shift to the right by 4 units. When we substitute \( x = 2 \), we get:\[ 4 - x = 4 - 2 = 2 \]Since there is a reflection, the transformation actually occurs by substituting \( x = -2 \). Thus, the horizontally transformed x-coordinate is \(-2\).

Step 3: Apply Vertical Scaling

The function \( y = \frac{1}{2} f(4-x) \) indicates a vertical scaling by a factor of \( \frac{1}{2} \). For the point \((x, y) = (2, -3)\), the \( y \)-value changes by multiplying by \( \frac{1}{2} \). This gives the new \( y \)-coordinate as:\[ y = \frac{1}{2} \times (-3) = -\frac{3}{2} \]

Step 4: Determine the Transformed Point

The transformed point is obtained by combining the new \( x \)-coordinate from Step 2 and the new \( y \)-coordinate from Step 3. Thus, the point on the graph of \( y = \frac{1}{2} f(4-x) \) is \((-2, -\frac{3}{2})\).

Key Concepts

Horizontal Reflection

When dealing with function transformations, a horizontal reflection is a transformation that flips the graph of a function over a vertical line. This can be thought of as a mirror reflecting the function's points. In the given function, the term

• \( 4-x \) can be rewritten as \(-1(x-4)\).

This indicates a horizontal reflection. This reflection occurs across a vertical line, in this case, the line \( x = 4 \). This means that every point \((x, y)\) on the original function is mirrored at an equivalent distance on the opposite side of the axis of reflection. For example, if a point was at \( x = 2 \), due to this reflection, it shifts to be the point that would be at \( x = -2 \), flipping over the vertical line crossing \( x = 4 \). It's like looking at the function from the other side of the mirror.

Vertical Scaling

Vertical scaling affects how a function stretches or compresses vertically. It is achieved by multiplying the function by a constant factor. In our case, the function is multiplied by

• \( \frac{1}{2} \).

This means every \( y \)-coordinate on the graph is reduced to half its original size. For instance, if the original point on the function \( y = f(x) \) was \((2, -3)\), after the transformation, the \( y \)-coordinate will become:

• \( y = \frac{1}{2} \times (-3) = -\frac{3}{2} \).

The graph appears compressed and shorter, closer to the \( x \)-axis. This change alters the function's steepness, making rising and falling of the graph appear less pronounced.

Horizontal Shift

A horizontal shift occurs when a function is moved left or right on the graph. For the given problem, the function modifies the input

• \( 4-x \).

This can be broken down as

• \(-1(x-4)\).

The part \(-4\) indicates a horizontal shift four units to the right compared with the original function \( f(x) \). Essentially, the entire graph of the function moves along the \( x \)-axis without the shape of the graph changing, apart from reflections or scalings. Hence, the point \((x, y)\) is adjusted horizontally, adjusting the input from \( x \) to \( -2 \), when substituting to find actual values.

Precalculus

Precalculus is often the gateway for understanding deeper mathematical concepts like calculus. This study area includes an array of essential topics like functions, transformations, and more. Understanding transformations, such as

• horizontal reflections,
• vertical scaling,
• and horizontal shifts,

plays a key role in precalculus. It equips students with a toolkit for analyzing and predicting the behavior of various functions. Mastering these concepts allows learners to see functional relationships, manipulate function representations, and prepare for the complexities of calculus. Precalculus is like crafting the basic language needed to express and understand the patterns you'll encounter in higher mathematics.