Problem 18 - 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. $$ y=\frac{4-f(3 x-1)}{7} $$

Step-by-Step Solution

Answer

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

Step 1: Understand Theorem 1.7

Theorem 1.7 pertains to transformations of functions involving translation, reflection, stretch, and compression. For a function transformation in the form of \( y = af(bx + c) + d \), the transformations include horizontal scaling, horizontal translation, reflection over the x-axis, vertical scaling, and vertical translation.

Step 2: Identify given point in terms of function

The known point \((2, -3)\) tells us that when \(x = 2\), \(f(2) = -3\). Therefore, a point \((2, -3)\) lies on the graph of \(y = f(x)\).

Step 3: Determine inside transformation effect

The transformation \( y = \frac{4 - f(3x - 1)}{7} \) indicates the substitution of \(3x - 1\) for \(x\) in the function \(f\). In terms of the function, this means a horizontal scaling by a factor of \(\frac{1}{3}\) and a horizontal shift to the right by \(\frac{1}{3}\).

Step 4: Solve for new x-value

Set \(3x - 1 = 2\) to find the new \(x\) that corresponds to the old \(x\). Solving this equation gives \(x = 1\).

Step 5: Apply vertical transformation

The function \( y = \frac{4 - f(3x - 1)}{7} \) has a vertical flip (reflection) due to the negative sign and a vertical compression by a factor of \(7\). Given \( f(2) = -3 \), the transformed value is \( y = \frac{4 - (-3)}{7} = \frac{7}{7} = 1 \).

Step 6: Write the new point

The new point on the graph of the transformed function based on the transformations identified is \((1, 1)\).

Key Concepts

Horizontal Scaling

When we talk about horizontal scaling in the world of functions, we're referring to how a function is stretched or compressed along the x-axis. Imagine a piece of stretchy tape you're pulling or pushing on both ends. That's similar to how horizontal scaling works on the graph of a function. In mathematical terms, if you see an expression like \( bx \) inside a function, it means the function is horizontally transformed.

• If \( b > 1 \), the function will compress horizontally, which makes it appear narrower.
• If \( 0 < b < 1 \), the function will stretch out, making it look wider.

In our example, we see \( 3x - 1 \). The coefficient of \( x \), which is 3, indicates a horizontal compression by a factor of 3 because \( b = 3 \), which is greater than 1. To adjust it back to its original scale, any point \( (x, y) \) on the transformed graph means that the input was scaled down; therefore, relate the input to its new placement with respect to 3. In this transformation, \( x \) gets squeezed into less space, thus affecting the arrangement and spacing of any points on the graph horizontally.

Vertical Translation

Vertical translation involves moving a graph up or down along the y-axis. It's like placing a piece of paper with your graph on it up or down on a table.
This is shown in the expression as a constant term added or subtracted from the function. The general form to understand is \( y = f(x) + d \), where \( d \) determines the direction of movement:

• \( d > 0 \) means the graph moves up.
• \( d < 0 \) means the graph moves down.

In the transformation equation \( y = \frac{4 - f(3x - 1)}{7} \), after reflecting and scaling, there's a downward movement if all operations result in a shift of terms. With the vertical compression by 7, we observe this shift in position eventually stabilizes directly influenced by the fraction's form, making solving the original equation more intuitive as you adjust by these vertical shifts.

Reflection Over the x-axis

Reflection over the x-axis means flipping a graph upside down. It's like looking in a mirror placed horizontally along the x-axis. Every point \( (x, y) \) on the graph gets mirrored to \( (x, -y) \).
This transformation is identified in an expression by a negative sign outside the function term, like \( -f(x) \). If the reflection is part of more complex operations, a careful step-by-step sign change keeps track of the flipping. For the example function \( y = \frac{4 - f(3x - 1)}{7} \), the negative sign in front of \( f(3x - 1) \) indicates such a reflection.
When reflecting, think about how each part or curve reappears flipped. The maximums become minimums, and the minimums become maximums, echoing symmetrical shapes across the x-axis without altering their horizontal positions. This requirement to change y-values' sign is central to completing any transformation that includes flipping.

Function Transformations

Function transformations describe all the actions taken to predictably change the graph of a function. By understanding these transformations, any graph can be adjusted to fit any required constraints or properties in analysis or practical applications.
The main transformations include:

• Vertical and horizontal translations, moving the function along the y-axis or x-axis, respectively.
• Horizontal and vertical scaling, stretching or compressing the function's appearance.
• Reflections across the x-axis or y-axis, flipping the function graphically.

In our transformed example \( y = \frac{4 - f(3x - 1)}{7} \), the function undergoes several transitions:

• Horizontal compression by a factor of 3 (horizontal scaling).
• Reflective flipping vertically over the x-axis (reflection).
• Vertical compression by a factor of 7 (vertical scaling).

Knowing how to read and apply these transformations allows for seamless graph manipulation and insights into behavior prediction in equations. Setting these steps in logical order ensures clarity.