Problem 73
Question
The graph of \(f(x)=10^{x}\) is reflected about the \(x\) -axis and shifted upward 7 units. What is the equation of the new function, \(g(x) ?\) State its -intercept, domain, and range.
Step-by-Step Solution
Verified Answer
The equation of the new function is \(g(x) = -10^x + 7\); its y-intercept is (0, 6), domain \((-
\infty, \, \infty)\), range \((-
\infty, 7)\).
1Step 1: Understand the Reflection
The reflection of the function around the x-axis means that we multiply the function by -1. Thus, reflecting the graph of \(f(x)=10^x\) about the x-axis, we get \(f(x) = -10^x\).
2Step 2: Apply Vertical Shift
After reflection, we shift the entire graph upwards by 7 units. This means we add 7 to the function: \(-10^x + 7\). Thus, the new function is \(g(x) = -10^x + 7\).
3Step 3: Find the y-intercept
To find the y-intercept, set \(x = 0\) in the equation \(g(x) = -10^x + 7\). We calculate \(g(0) = -10^0 + 7 = -1 + 7 = 6\). So, the y-intercept is \((0, 6)\).
4Step 4: Determine the Domain
The domain of \(g(x) = -10^x + 7\) is all real numbers, since there are no restrictions on \(x\) for exponential functions. So, the domain is \((-\infty, \, \infty)\).
5Step 5: Determine the Range
After reflecting \(f(x) = 10^x\) across the x-axis, it becomes \(g(x) = -10^x\), which has a range of \( (-\infty, 0) \). By shifting it upwards by 7 units, the range of \(g(x) = -10^x + 7\) becomes \((-\infty, 7)\).
Key Concepts
Reflection of FunctionsVertical ShiftsExponential FunctionsDomain and Range
Reflection of Functions
Reflecting functions is an important transformation that involves flipping the function over a specific axis. When we reflect a function about the x-axis, each point on the graph is transformed to the same distance on the opposite side of the x-axis.
In the case of the function \( f(x) = 10^x \), reflecting about the x-axis means multiplying the entire function by \(-1\). This will change the function into \(-10^x\). This transformation flips the exponential curve upside down, making it decrease as it moves left to right.
Remember, this reflection affects only the vertical aspect of the graph, while the x-values, or inputs, stay the same.
In the case of the function \( f(x) = 10^x \), reflecting about the x-axis means multiplying the entire function by \(-1\). This will change the function into \(-10^x\). This transformation flips the exponential curve upside down, making it decrease as it moves left to right.
Remember, this reflection affects only the vertical aspect of the graph, while the x-values, or inputs, stay the same.
Vertical Shifts
Vertical shifts are transformations that move the graph up or down on a coordinate plane. This shift is dependent on whether a constant is added or subtracted from the function.
For example, if you want to shift the reflected graph of \(-10^x\) upwards by 7 units, you simply add 7 to the function. Thus, it modifies the function to \( g(x) = -10^x + 7 \).
This adjustment moves every point on the graph 7 units higher, while maintaining the same reflective shape. It's important to note that this shift doesn't affect the domain of the function but does alter the range.
For example, if you want to shift the reflected graph of \(-10^x\) upwards by 7 units, you simply add 7 to the function. Thus, it modifies the function to \( g(x) = -10^x + 7 \).
This adjustment moves every point on the graph 7 units higher, while maintaining the same reflective shape. It's important to note that this shift doesn't affect the domain of the function but does alter the range.
Exponential Functions
Exponential functions are characterized by the form \( f(x) = a^x \) where \( a \) is a positive constant. These functions grow or decay at a rate proportional to their current value.
Specifically, \( f(x) = 10^x \) represents an exponential function where the base is 10. Such functions are powerful as they model many natural processes like population growth and radioactive decay.
However, transformations such as reflections or vertical shifts can affect the behavior and appearance of these functions. While the basic form and consistency remain, transformations can impact their graphical representation and intersections with axes.
Specifically, \( f(x) = 10^x \) represents an exponential function where the base is 10. Such functions are powerful as they model many natural processes like population growth and radioactive decay.
However, transformations such as reflections or vertical shifts can affect the behavior and appearance of these functions. While the basic form and consistency remain, transformations can impact their graphical representation and intersections with axes.
Domain and Range
In the context of functions, the domain refers to the set of all possible input values (x-values) that will produce a valid output. For exponential functions such as \( g(x) = -10^x + 7 \), the domain is all real numbers, \((-\infty, \infty)\), because there are no restrictions on the exponent of a number.
On the other hand, the range signifies all possible output values (y-values) that the function can produce. Reflections and shifts critically alter the range. After reflecting \( 10^x \) and shifting it up, the range of \( g(x) = -10^x + 7 \) is \((-\infty, 7)\). This means that no value for \( g(x) \) can exceed 7, adhering to both transformations performed on the original function.
On the other hand, the range signifies all possible output values (y-values) that the function can produce. Reflections and shifts critically alter the range. After reflecting \( 10^x \) and shifting it up, the range of \( g(x) = -10^x + 7 \) is \((-\infty, 7)\). This means that no value for \( g(x) \) can exceed 7, adhering to both transformations performed on the original function.
Other exercises in this chapter
Problem 71
The graph of \(f(x)=3^{x}\) is reflected about the \(y\) -axis and stretched vertically by a factor of \(4 .\) What is the equation of the new function, \(g(x)
View solution Problem 72
The graph of \(f(x)=\left(\frac{1}{2}\right)^{-x}\) is reflected about the \(y\) -axis and compressed vertically by a factor of \(\frac{1}{5}\) . What is the eq
View solution Problem 74
The graph of \(f(x)=(1.68)^{x}\) is shifted right 3 units, stretched by a factor of \(2,\) reflected about the \(x\) -axis, and then shifted downward 3 units. W
View solution Problem 75
The graph of \(f(x)=-\frac{1}{2}\left(\frac{1}{4}\right)^{x-2}+4\) is shifted left 2 units, stretched vertically by a factor of \(4,\) reflected about the \(x\)
View solution