Problem 83
Question
The point (2,4) on the graph of \(f(x)=x^{2}\) has been shifted horizontally to the point \((-3,4) .\) Identify the shift and write a new function \(g(x)\) in terms of \(f(x)\).
Step-by-Step Solution
Verified Answer
The shift is 5 units to the left. The new function, \( g(x) \), in terms of \( f(x) \) is \( g(x)= (x+5)^{2} \).
1Step 1: Determine the Direction and Size of the Shift
The original point is (2,4) and the new point after the shift is (-3,4). The y-values remain unchanged (4), signaling a horizontal shift occurred. The x-value moved from 2 to -3, an apparent leftward shift of 5 units.
2Step 2: Write the New Function in Terms of the Old One
After determining the shift is 5 units to the left, this can be expressed mathematically in the new function \( g(x) \) by replacing every 'x' in the original function \( f(x) \) with '\( x+5 \)' . So if \( f(x)=x^{2}\), then \( g(x)= (x+5)^{2}\), representing a function identical to \( f(x) \), but shifted 5 units to the left.
Key Concepts
Transformation of FunctionsWriting FunctionsFunction Notation
Transformation of Functions
When studying mathematics, understanding the transformation of functions is a critical concept that allows alterations to the graph of a function without changing its core attributes. In the context of our exercise, a horizontal shift is a type of transformation that slides a graph left or right along the x-axis.
To visualize a horizontal shift, imagine the entire graph of a function is being nudged horizontally in a particular direction. The y-values of the points on the graph remain constant since the shift is parallel to the x-axis. In the accompanying exercise, the horizontal shift is determined by looking at the change in the x-values, while the y-values stay the same. Here, the function moved from (2,4) to (-3,4), indicating a leftward shift. This transformation is like picking up the whole graph and moving it 5 units to the left, corresponding to the x-value moving from 2 to -3.
To express a horizontal shift mathematically, we add or subtract a number to the x input of the function. If we’re moving to the right, we subtract from x, and if we’re moving to the left, we add to x. In our scenario, we are dealing with a leftward shift, hence we add 5 to the x, which gives us the transformation to apply: g(x) = f(x + 5), where f(x) is the original function.
To visualize a horizontal shift, imagine the entire graph of a function is being nudged horizontally in a particular direction. The y-values of the points on the graph remain constant since the shift is parallel to the x-axis. In the accompanying exercise, the horizontal shift is determined by looking at the change in the x-values, while the y-values stay the same. Here, the function moved from (2,4) to (-3,4), indicating a leftward shift. This transformation is like picking up the whole graph and moving it 5 units to the left, corresponding to the x-value moving from 2 to -3.
To express a horizontal shift mathematically, we add or subtract a number to the x input of the function. If we’re moving to the right, we subtract from x, and if we’re moving to the left, we add to x. In our scenario, we are dealing with a leftward shift, hence we add 5 to the x, which gives us the transformation to apply: g(x) = f(x + 5), where f(x) is the original function.
Writing Functions
Writing functions effectively is a fundamental skill for expressing mathematical ideas with precision. In general, a function is a relationship that assigns exactly one output to each input. The systematic way to write functions involves using function notation, which is crucial for clarity and communication in mathematics.
When composing the new function after a transformation, it's important to maintain the original function's structure while incorporating the changes. In the case of horizontal shifts, the input of the original function is adjusted to reflect the shift. For our exercise, the transformation dictates that we need to replace every x in the original function with (x + 5) to account for the 5 unit shift to the left. Hence, if our original function is f(x) = x^2, writing the new function involves rewriting it with the adjustment: g(x) = (x + 5)^2.
The correctness of writing functions also lies in understanding what modifications are required. Here, no change is needed for the exponent or coefficients because the shape and orientation of the graph remain the same; it's only the position that changes.
When composing the new function after a transformation, it's important to maintain the original function's structure while incorporating the changes. In the case of horizontal shifts, the input of the original function is adjusted to reflect the shift. For our exercise, the transformation dictates that we need to replace every x in the original function with (x + 5) to account for the 5 unit shift to the left. Hence, if our original function is f(x) = x^2, writing the new function involves rewriting it with the adjustment: g(x) = (x + 5)^2.
The correctness of writing functions also lies in understanding what modifications are required. Here, no change is needed for the exponent or coefficients because the shape and orientation of the graph remain the same; it's only the position that changes.
Function Notation
Function notation is the formal way to communicate the definition of a function and its corresponding domain and range. It is an efficient system that uses symbols to express the relationship between the input (usually represented as x) and the output. With function notation, we can define, evaluate, and perform operations on functions effectively.
In our exercise, f(x) = x^2 is the function notation indicating that the function f takes an input x and squares it to produce the output. After performing a horizontal shift, we use function notation to define the new function g. The notation g(x) is used to state that for every x, there is an output that follows the rule defined by g. In this case, every x in the original function is replaced with (x + 5) due to the shift, leading to the notation g(x) = (x + 5)^2.
This notation not only specifies the rule by which g operates but also provides a clear and concrete method to identify the relationship between the input and output of the new, transformed function.
In our exercise, f(x) = x^2 is the function notation indicating that the function f takes an input x and squares it to produce the output. After performing a horizontal shift, we use function notation to define the new function g. The notation g(x) is used to state that for every x, there is an output that follows the rule defined by g. In this case, every x in the original function is replaced with (x + 5) due to the shift, leading to the notation g(x) = (x + 5)^2.
This notation not only specifies the rule by which g operates but also provides a clear and concrete method to identify the relationship between the input and output of the new, transformed function.
Other exercises in this chapter
Problem 82
Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have two real zeros?
View solution Problem 83
A rectangular sandbox is to be enclosed with a fence on three of its sides and a brick wall on the fourth side. If 24 feet of fencing material is available, wha
View solution Problem 83
In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{x^{2}+1
View solution Problem 83
Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have one real zero?
View solution