Problem 90 - College Algebra with Modeling and Visualization

Question

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f\left(-\frac{1}{2} x\right)\)

Step-by-Step Solution

Answer

The points are (24, 6), (0, 8), and (-16, -4).

Step 1: Understand the Transformation

The function \( g(x) = f\left(-\frac{1}{2}x\right) \) involves a horizontal transformation of the original function \( f(x) \). Specifically, it consists of a horizontal reflection across the y-axis (due to the negative sign) and a horizontal stretch by a factor of 2 (due to the \( \frac{1}{2} \) factor).

Step 2: Determine New Points for Transformation

For each given point \((x, y)\) on \( f(x) \), the corresponding point on \( g(x) \) will be \(( -2x, y)\). This is because the transformation requires multiplying the x-coordinate by -2.

Step 3: Apply Transformation to Each Point

- Transform \((-12, 6)\): - The new x-coordinate is \( -2(-12) = 24 \). - So the point becomes \((24, 6)\).- Transform \((0, 8)\): - The new x-coordinate is \( -2(0) = 0 \). - So the point becomes \((0, 8)\).- Transform \((8, -4)\): - The new x-coordinate is \( -2(8) = -16 \). - So the point becomes \((-16, -4)\).

Key Concepts

Horizontal Reflection

When talking about a horizontal reflection in the context of function transformations, we're discussing flipping the graph of the function across the y-axis. This type of transformation happens when the x in the function is multiplied by a negative number.
For instance, in the transformation from \(f(x)\) to \(g(x) = f(-\frac{1}{2}x)\), the negative sign in front of the \(\frac{1}{2}\) inside the function causes a horizontal reflection.

• This reflection means that each point \((x, y)\) on \(f(x)\) is moved to \((-x, y)\) in its transformed version before any additional transformations are applied.

In the exercise, this was why, for the point \((-12, 6)\), only the x-coordinate became positive after multiplying with -1, giving us \((24, 6)\) as part of the further transformation processes.

Horizontal Stretch

A horizontal stretch affects the shape of a graph by pulling it away from or compressing it towards the y-axis. This happens when each x-coordinate of the function is divided by or multiplied by a constant factor.
In the exercise, we see this concept with the factor of \(\frac{1}{2}\) within the function \(g(x)=f\left(-\frac{1}{2}x\right)\).

• Instead of compressing to half the original width, a horizontal stretch by a factor of 2 is happening.
• This is because inversely, multiplying the x-coordinate by -2 (as shown in the transformation process), effectively stretches it.

Consequently, while reflecting, each originally given point's x-coordinate is multiplied by -2. So for the points on \(f(x)\), such as \((8, -4)\), the x is doubled in its negative form, creating \((-16, -4)\) on \(g(x)\).

Coordinate Transformation

Coordinate transformation in this scenario involves changing the coordinates of points from one function to another by applying specified rules.
The given transformation \(g(x) = f\left(-\frac{1}{2}x\right)\) instructs on handling each point given on \(f(x)\).

• First, multiply the x-coordinate by -2 as explained in combining horizontal reflection and stretch.
• Notice here, the y-coordinate remains unchanged and only the x undergoes the alteration.

Thus, applying this coordinate transformation rule to a point like \((0, 8)\) maintains its position since its x-related component transitions to \(0\). The point remains \((0, 8)\) on \(g(x)\). This transformation retains the shape vertically while adjusting horizontally according to the specified operations.