Problem 56 - College Algebra

Question

describe how the graph of each function is a transformation of the graph of the original function \(f.\) $$g(x)=6 f(x)$$

Step-by-Step Solution

Answer

The graph of \( g(x) = 6f(x) \) is a vertical stretch of \( f(x) \) by a factor of 6.

Step 1: Understanding the Base Function

Let's start by understanding that the function given is a transformation of some original function denoted as \( f(x) \). We do not have the explicit form of \( f(x) \), but we know that \( g(x) = 6f(x) \) involves multiplying the original function by 6.

Step 2: Identify the Transformation Type

Observe that the transformation involves multiplying \( f(x) \) by a constant (6). This tells us that the transformation is a vertical scaling or stretching.

Step 3: Determine the Transformation Effect

Since the constant is greater than 1, the graph of \( f(x) \) is stretched vertically by a factor of 6. Each point \((x, f(x))\) on the graph of \( f(x) \) will transform into \((x, 6f(x))\) on the graph of \( g(x) \). This means that every y-coordinate becomes 6 times larger, making the graph taller.

Step 4: Graphical Interpretation

Visualize that if the original function \( f(x) \) was a curve on the graph, the function \( g(x) \) will have the same shape but will be stretched upward. For example, if \( f(x) \) crosses the y-axis at 1, \( g(x) \) will cross at 6.

Key Concepts

Vertical Scaling

Vertical scaling is a common form of transformation for functions. When you multiply a function by a constant, say 6, you are effectively "stretching" or "compressing" the graph vertically. If the constant is greater than 1, like in our case with 6, the graph stretches upwards.Vertical scaling affects the output of the function. Instead of changing its horizontal placement, it makes the entire graph taller or shorter. In the function transformation described as \(g(x) = 6f(x)\):

• The original function \(f(x)\) is vertically stretched by a factor of 6.
• This means for every point on the graph, the \(y\)-value of \(f(x)\) is multiplied by 6.

This simple multiplication affects the steepness of any curves or lines that exist in the function. It enlarges the distance every point moves away from the x-axis, giving a dramatic change when you look at the graph after transformation.

Graph Interpretation

Interpreting graphs after transformations can be easier once you understand what operations like vertical scaling do. The visual effect is that the shape of the graph remains identical to the original; however, its height changes. When looking at the graph of \(f(x)\) versus \(g(x) = 6f(x)\):

• The same critical points such as intercepts, peaks, and valleys will occur with increased height.
• For instance, if \(f(x)\) has a point at \((x_0, y_0)\), then \(g(x)\) will have a corresponding point at \((x_0, 6y_0)\).

This implies if the original function \(f(x)\) were plotted, \(g(x)\) would look like it's the same graph but blown up vertically. It’s important to remember that the x-intercepts of the function remain unchanged, as stretching happens only vertically.

Y-coordinate Transformation

A key concept in function transformations, particularly with vertical scaling, is how the y-coordinates transform. In our equation \(g(x) = 6f(x)\), every y-coordinate from \(f(x)\), denoted as \(y = f(x)\), is transformed by multiplying it by 6 in \(g(x)\). This means every output value of the original function increases six times. Consider these effects in scenarios:

• If \(f(x) = 2\), then \(g(x) = 2 \times 6 = 12\).
• If \(f(x)\) was negative, the effect of multiplying by 6 would similarly amplify the negativity, e.g., \(-1 \to (-1) \times 6 = -6\).

This multiplication affects only the vertical dimension and is what differentiates it from horizontal transformations where x-coordinates would change. Ultimately, vertical scaling through y-coordinate transformation not only makes graphs taller or shorter but can also accentuate certain characteristics within the graph’s dynamics.