Problem 52
Question
List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$f(x)=\ln x-7$$
Step-by-Step Solution
Verified Answer
Answer: The transformation is a vertical shift down 7 units.
1Step 1: Identify the Base Function
We are given the base function:
$$g(x) = \ln x$$
We need to find the transformations from the base function to the derived function:
$$f(x)=\ln x - 7$$
2Step 2: Identify the Vertical Shift
The derived function \(f(x) = \ln x - 7\) has a constant term, which is -7. This indicates that the graph of \(f(x)\) is the graph of \(g(x)\) shifted vertically down by 7 units. To write this as a transformation, we can say:
- Vertical Shift: Down 7 units
There are no other changes in the function \(f(x) = \ln x - 7\) compared to the base function \(g(x) = \ln x\). Therefore, the only transformation required is the vertical downward shift of 7 units.
Key Concepts
Base FunctionVertical ShiftNatural Logarithm GraphTransformation of Functions
Base Function
Understanding the concept of a base function is crucial when studying graph transformations in precalculus. In essence, a base function is the simplest form of a more complex function and serves as the starting point for any transformation. For instance, if we consider the function g(x) = ln(x), this represents the natural logarithm function without any modifications or shifts. This is the foundational graph we use for comparison when identifying changes in its shape, position, or orientation.
When facing an exercise that involves modifying this function, we start by looking at this initial, unaltered form and then apply transformations to obtain the new function. It's crucial to have a strong grasp of the characteristics of the base function, as this knowledge forms the bedrock for understanding the effects that each transformation will have on the graph of the function.
When facing an exercise that involves modifying this function, we start by looking at this initial, unaltered form and then apply transformations to obtain the new function. It's crucial to have a strong grasp of the characteristics of the base function, as this knowledge forms the bedrock for understanding the effects that each transformation will have on the graph of the function.
Vertical Shift
The term 'vertical shift' is used to describe a specific type of graph transformation that moves the graph up or down along the y-axis without altering its shape. This movement is one of the simplest transformations to identify and apply. It occurs when a constant is either added to or subtracted from the base function. In the context of our exercise, the presence of -7 in the function f(x) = ln(x) - 7 signifies a vertical shift.
To apply a vertical shift, we simply take every point on the base graph and move it up or down the y-axis by the constant value. In the case of f(x), adding -7 to the base function g(x) means we're shifting the graph downwards by 7 units. This downward shift does not change the graph's shape or horizontal location; it just translates the entire graph down, as if sliding it lower on a sheet of graph paper.
To apply a vertical shift, we simply take every point on the base graph and move it up or down the y-axis by the constant value. In the case of f(x), adding -7 to the base function g(x) means we're shifting the graph downwards by 7 units. This downward shift does not change the graph's shape or horizontal location; it just translates the entire graph down, as if sliding it lower on a sheet of graph paper.
Natural Logarithm Graph
The graph of the natural logarithm function, typically written as g(x) = ln(x), has distinct characteristics making it uniquely recognizable. Firstly, it's defined for all positive values of x, approaching infinity as x grows larger and approaching negative infinity as x approaches zero. The natural logarithm graph is not defined for x less than or equal to zero and is characterized by a vertical asymptote along the y-axis.
The graph is continuously increasing, but the rate of increase slows as x becomes larger, known as logarithmic growth. The natural logarithm graph intersects the x-axis at x=1, where the value of ln(1) is zero. Understanding this base graph is instrumental for students to grasp how alterations like vertical shifts will impact its overall appearance.
The graph is continuously increasing, but the rate of increase slows as x becomes larger, known as logarithmic growth. The natural logarithm graph intersects the x-axis at x=1, where the value of ln(1) is zero. Understanding this base graph is instrumental for students to grasp how alterations like vertical shifts will impact its overall appearance.
Transformation of Functions
In precalculus, a transformation of functions involves altering the base graph in various ways to produce a new graph that has been stretched, compressed, shifted, or reflected. There are several fundamental types of transformations:
In our exercise example, we are focused on a vertical shift, which is just one facet of function transformation. By learning about these transformations and how they affect the graph, students gain a versatile toolset for quickly and accurately sketching the graphs of complex functions. This knowledge is underpinned by an understanding of how shifts, stretches, compressions, and reflections visually modify the shape and position of the base function's graph on a coordinate plane.
- Vertical and horizontal shifts
- Vertical and horizontal stretches and compressions
- Reflections across the axes
In our exercise example, we are focused on a vertical shift, which is just one facet of function transformation. By learning about these transformations and how they affect the graph, students gain a versatile toolset for quickly and accurately sketching the graphs of complex functions. This knowledge is underpinned by an understanding of how shifts, stretches, compressions, and reflections visually modify the shape and position of the base function's graph on a coordinate plane.
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