Problem 56
Question
Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{10} x, \quad g(x)=\log _{10}(x+7)$$
Step-by-Step Solution
Verified Answer
The transformation of the graph of \(f(x) = log_{10} x\) to the graph of \(g(x) = log_{10} (x + 7)\) is a shift 7 units to the left.
1Step 1: Identify the transformation type
Comparing the two functions \(f(x) = log_{10} x\) and \(g(x) = log_{10} (x + 7)\) together, one can see that a transformation has taken place. The X variable in the function \(g(x)\) has been augmented by a positive 7 (\(x + 7\)). This indicates a horizontal shift, or translation, of the graph.
2Step 2: Determine the direction and magnitude of the shift
In general, in the function notation \(g(x) = f(x + h)\), the \(+h\) inside the function represents a horizontal shift in the opposite direction of the sign. So, \(x + 7\) in this case represents a shift 7 units to the left.
3Step 3: Summarize the transformation
So, the transformation of the graph of \(f(x) = log_{10} x\) that yields the graph of \(g(x) = log_{10} (x + 7)\) is a shift 7 units to the left. This is the transformation from \(f\) to \(g\).
Key Concepts
Function TransformationHorizontal ShiftGraphing Logarithmic Functions
Function Transformation
Function transformation involves changing the position, shape, or size of a graph in a predictable way. Logarithmic functions, like any other mathematical functions, can undergo these transformations. A transformation can include translations (shifts), reflections, scaling (stretching or compressing), and rotations.
In the case of the original functions, we are dealing with a logarithmic function transformation. By comparing the base function \(f(x) = \log_{10}x\) to the transformed function \(g(x) = \log_{10}(x+7)\), we identify that the transformation applied is a translation. Specifically, transformation in logarithmic functions often involves changing the input variable \(x\) to \(x + k\) or \(x - k\), where \(k\) modifies the position of the graph along the x-axis.
This ability to transform functions allows mathematicians and students to easily visualize and understand the impact of various alterations on the function's graph, making them fundamental in the study of graph behaviors and properties.
In the case of the original functions, we are dealing with a logarithmic function transformation. By comparing the base function \(f(x) = \log_{10}x\) to the transformed function \(g(x) = \log_{10}(x+7)\), we identify that the transformation applied is a translation. Specifically, transformation in logarithmic functions often involves changing the input variable \(x\) to \(x + k\) or \(x - k\), where \(k\) modifies the position of the graph along the x-axis.
This ability to transform functions allows mathematicians and students to easily visualize and understand the impact of various alterations on the function's graph, making them fundamental in the study of graph behaviors and properties.
Horizontal Shift
A horizontal shift involves moving the entire graph of a function left or right on the Cartesian plane. This type of transformation is pivotal for understanding how functions can be altered and visualized differently.
In our example, we examine the expression \(x+7\) inside the logarithmic function \(g(x) = \log_{10}(x+7)\). According to transformation rules, the term \(+7\) indicates a shift to the left by 7 units, contrary to the positive sign suggesting a right shift. This is because the formula \(g(x) = f(x + h)\) shifts the graph \(\left|h\right|\) units in the opposite direction of the sign inside the function.
Properly understanding horizontal shifts prepares students to anticipate the effects of algebraic manipulations on the graph's direction and magnitude of movement, which is crucial for plotting and graphing accuracy.
In our example, we examine the expression \(x+7\) inside the logarithmic function \(g(x) = \log_{10}(x+7)\). According to transformation rules, the term \(+7\) indicates a shift to the left by 7 units, contrary to the positive sign suggesting a right shift. This is because the formula \(g(x) = f(x + h)\) shifts the graph \(\left|h\right|\) units in the opposite direction of the sign inside the function.
Properly understanding horizontal shifts prepares students to anticipate the effects of algebraic manipulations on the graph's direction and magnitude of movement, which is crucial for plotting and graphing accuracy.
Graphing Logarithmic Functions
Graphing logarithmic functions can be more challenging than linear functions, yet it is essential to master this skill to visualize different mathematical functions. Logarithmic functions typically have the form \(f(x) = \log_b(x)\), where \(b\) is the base of the logarithm.
When plotting \(f(x) = \log_{10} x\), the graph passes through the point (1, 0) because \(\log_{10}(1) = 0\). It approaches the vertical axis (y-axis) but never touches it, reflecting its asymptotic nature.
With the transformation seen in \(g(x) = \log_{10}(x+7)\), the graph is shifted 7 units to the left. Therefore, points on the graph of \(f(x)\) also shift 7 units left, changing the intercept to (-6, 0). When graphing these functions, it's important to note log base properties and asymptotes which define their behavior clearly.
Understanding how to plot these changes enhances one's ability to reason mathematically and graphically, showcasing the power of function transformations in altering graphs' orientations without altering their overall shape.
When plotting \(f(x) = \log_{10} x\), the graph passes through the point (1, 0) because \(\log_{10}(1) = 0\). It approaches the vertical axis (y-axis) but never touches it, reflecting its asymptotic nature.
With the transformation seen in \(g(x) = \log_{10}(x+7)\), the graph is shifted 7 units to the left. Therefore, points on the graph of \(f(x)\) also shift 7 units left, changing the intercept to (-6, 0). When graphing these functions, it's important to note log base properties and asymptotes which define their behavior clearly.
Understanding how to plot these changes enhances one's ability to reason mathematically and graphically, showcasing the power of function transformations in altering graphs' orientations without altering their overall shape.
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Problem 56
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