Write An Equation For The Transformed Logarithm Shown Below

Write an Equation for the Transformed Logarithm Shown Below

This article delves into the process of deriving the equation for a transformed logarithmic function, given its visual representation. We'll explore various transformations, including vertical and horizontal shifts, stretches, and reflections, and how these affect the parent logarithmic function, y = log_b(x). Understanding these transformations is crucial for accurately modeling real-world phenomena that exhibit logarithmic behavior, such as earthquake magnitudes, sound intensity, and population growth.

Understanding the Parent Logarithmic Function: y = log_b(x)

Before tackling transformations, let's solidify our understanding of the base logarithmic function, y = log_b(x). Here, 'b' represents the base of the logarithm (b > 0, b ≠ 1), and 'x' is the argument. This function has a vertical asymptote at x = 0 (the y-axis), meaning the function approaches but never touches this line. The x-intercept is at (1, 0). The function's behavior depends heavily on the base 'b':

• If b > 1: The function is increasing. As x increases, y increases, albeit slowly.
• If 0 < b < 1: The function is decreasing. As x increases, y decreases.

Common Logarithmic Transformations

Transformations modify the parent function, shifting, stretching, or reflecting it. Let's examine the common transformations and how they alter the equation:

1. Vertical Shift:

A vertical shift moves the graph up or down along the y-axis. Adding a constant 'k' to the function shifts it vertically:

y = log_b(x) + k

• k > 0: Shifts the graph up by 'k' units.
• k < 0: Shifts the graph down by 'k' units.

The asymptote remains at x = 0.

2. Horizontal Shift:

A horizontal shift moves the graph left or right along the x-axis. Subtracting a constant 'h' from the argument shifts it horizontally:

y = log_b(x - h)

• h > 0: Shifts the graph right by 'h' units. The asymptote moves to x = h.
• h < 0: Shifts the graph left by 'h' units. The asymptote moves to x = h.

3. Vertical Stretch/Compression:

A vertical stretch or compression scales the graph vertically. Multiplying the function by a constant 'a' affects the vertical scaling:

y = a * log_b(x)

• |a| > 1: Stretches the graph vertically.
• 0 < |a| < 1: Compresses the graph vertically.
• a < 0: Reflects the graph across the x-axis (inverts the function).

4. Horizontal Stretch/Compression:

A horizontal stretch or compression scales the graph horizontally. This is achieved by multiplying the argument by a constant 'c':

y = log_b(cx)

• 0 < |c| < 1: Stretches the graph horizontally.
• |c| > 1: Compresses the graph horizontally.
• c < 0: Reflects the graph across the y-axis.

Combining Transformations

Real-world logarithmic functions often involve multiple transformations. The general form incorporating all these transformations is:

y = a * log_b(c(x - h)) + k

This equation encapsulates all possible combinations of vertical and horizontal shifts, stretches, compressions, and reflections. Remember the order of operations: parentheses (horizontal shift), then multiplication (horizontal stretch/compression), then the logarithm itself, then multiplication by 'a' (vertical stretch/compression), and finally the addition of 'k' (vertical shift).

Determining the Equation from a Graph: A Step-by-Step Approach

Let's assume you are given a graph of a transformed logarithmic function. To determine its equation, follow these steps:

1. Identify the Asymptote: Locate the vertical asymptote. This gives you the value of 'h'. The asymptote is at x = h.

2. Identify the Base (b): Determine the base of the logarithm. This might be explicitly stated, or you may need to deduce it based on the graph's behavior. If two points are given, you can use the definition of the logarithm to solve for the base. For example, if the graph passes through (1+h, k) and (b+h, k+1), then you can solve the equation alog_b(c(1+h-h)) + k = k and alog_b(c(b+h-h)) +k = k+1 for a,b,c.

3. Determine the Vertical Shift (k): Find the y-coordinate where x = 1+h. This y-coordinate is equal to k.

4. Find a Point (x, y): Select a point on the graph that is clearly visible and not too close to the asymptote.

5. Solve for 'a' and 'c': Substitute the values of 'h', 'k', and the chosen point (x,y) into the general equation. This will give you two equations with two unknowns, 'a' and 'c'. Solve this system of equations to find 'a' and 'c'. Note that if the graph is reflected, 'a' or 'c' will be negative.

6. Write the Final Equation: Substitute the values of 'a', 'b', 'c', 'h', and 'k' into the general equation: y = a * log_b(c(x - h)) + k.

Example: Determining the Equation of a Transformed Logarithm

Let's say a graph shows a logarithmic function with a vertical asymptote at x = 2, passing through the points (3, 0) and (4, 1), and seemingly following the general shape of a base 2 logarithm.

1. Asymptote: x = 2, so h = 2.
2. Base: We assume b = 2 (from the shape).
3. Vertical Shift: The graph passes through (3, 0). If b=2, then 0= alog₂(c(3-2)) + k. This suggests k=0.
4. Points: We have (3,0) and (4,1).
5. Solving for 'a' and 'c':
   • Using (3, 0): 0 = a * log₂(c(3 - 2)) +0 => 0 = a * log₂(c)
   • Using (4, 1): 1 = a * log₂(c(4 - 2)) + 0 => 1 = a * log₂(2c) If we assume a=1, then log₂(c)=0, which implies c=1. Substituting into the second equation, 1= log₂(2), which is true.
6. Final Equation: y = log₂(x - 2)

Therefore, the equation of the transformed logarithmic function is y = log₂(x - 2).

Conclusion

Determining the equation of a transformed logarithmic function from its graph requires a systematic approach. By understanding the impact of vertical and horizontal shifts, stretches, compressions, and reflections, and by carefully analyzing the graph's key features, you can accurately model the function's behavior using the general equation: y = a * log_b(c(x - h)) + k. This skill is essential for modeling various real-world phenomena and for solving problems in mathematics, science, and engineering that involve logarithmic relationships. Remember to always check your equation by plugging in known points from the graph to ensure accuracy. Practice with various examples to master this important skill in mathematical modeling.