Problem 89 - 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 \(y=g(x)\). \(g(x)=f(-2 x)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the transformation

The function \(g(x) = f(-2x)\) indicates a horizontal scaling of the function \(f\) by a factor of \(\frac{1}{2}\) and reflection about the y-axis. This means for any point \((a, b)\) on \(f(x)\), the corresponding point on \(g(x)\) is \((-\frac{a}{2}, b)\).

Step 2: Apply the transformation to the first point

Take the first point \((-12, 6)\) from \(f(x)\). To find the corresponding point on \(g(x)\), calculate \((-\frac{-12}{2}, 6)\). This simplifies to \( (6, 6)\).

Step 3: Apply the transformation to the second point

Take the second point \((0, 8)\) from \(f(x)\). Apply the transformation to get \((-\frac{0}{2}, 8)\), which is \((0, 8)\). This point remains unchanged after transformation.

Step 4: Apply the transformation to the third point

Use the point \((8, -4)\) from \(f(x)\). Transform it using \((-\frac{8}{2}, -4)\), which results in \((-4, -4)\).

Step 5: List the transformed points

The three transformed points that lie on the graph of \(g(x)\) are \((6, 6)\), \((0, 8)\), and \((-4, -4)\).

Key Concepts

Horizontal Scaling

When we talk about horizontal scaling, we refer to stretching or compressing the graph of a function horizontally. This transformation affects the x-values of the points on the graph. For our function transformation from \( f(x) \) to \( g(x) = f(-2x) \), we apply a horizontal scaling by a factor of \( \frac{1}{2} \).
This means every x-value on the original function \( f(x) \) is halved for the new function \( g(x) \). For example, if you have a point \((a, b)\) on the graph of \( f(x) \), the transformed point will have the x-coordinate \( -\frac{a}{2} \). The y-coordinate does not change.

Horizontal scaling by \( \frac{1}{2} \) makes the graph appear compressed towards the y-axis. Similarly, if the factor were greater than one, it would stretch away from the y-axis.
Keep in mind:

• A scaling factor between 0 and 1 compresses the graph.
• A scaling factor greater than 1 stretches the graph.
• The sign of the factor affects the direction (as we see in the reflection process).

Reflection about the y-axis

Reflection about the y-axis is another common transformation in coordinate geometry. When an expression like \( g(x) = f(-2x) \) includes a negative sign, it means the graph of \( f(x) \) is reflected over the y-axis in addition to any scaling.
To visualize this, picture flipping the graph of the function from right to left across the y-axis. For a point \((a, b)\) on \( f(x) \), reflecting about the y-axis would put the point at \((-a, b)\). When combined with horizontal scaling by a factor of \( \frac{1}{2} \), this transformation results in \( (-\frac{a}{2}, b) \).

This alteration is crucial in observing how the graph's orientation changes:

• This reflection doesn't affect the y-coordinates; it only impacts the x-coordinates.
• The y-axis acts as a mirror, reflecting each point horizontally.
• This transformation can change the direction of any symmetry the graph might have.

Graph of a Function

The graph of a function is a visual representation of all the possible solutions for the function. It allows us to understand the function's behavior over its domain. For any function \( f(x) \), the graph can provide insights into the function's properties such as where it increases or decreases, its intercepts, and its overall shape.
In our exercise, we started with the graph of \( f(x) \) defined by the points \((-12, 6)\), \((0, 8)\), and \((8, -4)\). These points were transformed using horizontal scaling and reflection to create the graph of \( g(x) = f(-2x) \).

Graphs are a powerful tool in mathematics as they:

• Make it easier to visualize function transformations.
• Allow us to see symmetry and other function characteristics directly.
• Help us predict other values for the function not explicitly given.

When we apply transformations, the graph helps confirm our calculations. For example, the point \((-12, 6)\) on \( f(x) \) transforms to \((6, 6)\) on \( g(x) \) showing us both the effect of horizontal scaling and reflection.