Question

The graph of the function \(y=x^{2}\) is translated 4 units to the left and 5 units up to form the transformed function \(y=g(x)\). a) Determine the equation of the function \(y=g(x)\) b) What are the domain and range of the image function? c) How could you use the description of the translation of the function \(y=x^{2}\) to determine the domain and range of the image function?

Short answer

a) \( y = g(x) = (x+4)^2 + 5 \)b) Domain: \( (-\infty, \infty) \) Range: \( [5, \infty) \)c) The description confirms the domain remains \( (-\infty, \infty) \); the range shifts to \( [5, \infty) \) due to the vertical translation.

Step-by-step solution

Understand the Transformation

The problem describes two translations: moving the graph 4 units to the left and 5 units up. Translations affect the function's equation as follows:- To move a function 4 units to the left, replace every occurrence of x with (x+4) in the function.- To move a function 5 units up, add 5 to the entire function.

Apply the Horizontal Translation

Start with the original function: \[ y = x^2 \] To translate it 4 units to the left, replace x with (x+4):\[ y = (x+4)^2 \]

Apply the Vertical Translation

Now take the horizontally translated function \( y = (x+4)^2 \) and translate it 5 units up by adding 5:\[ y = (x+4)^2 + 5 \] Thus, the transformed function is:\[ y = g(x) = (x+4)^2 + 5 \]

Determine the Domain

The domain of the original function \( y = x^2 \) is all real numbers, denoted by \( (-\infty, \infty) \). Translations do not affect the domain, so the domain of \( y = g(x) \) remains:\[ (-\infty, \infty) \]

Determine the Range

The range of the original function \( y = x^2 \) is \( [0, \infty) \) because \( x^2 \) is never negative and starts from 0. Translating the function 5 units up shifts the entire range up by 5. Therefore, the range of \( y = g(x) \) is:\[ [5, \infty) \]

Use the Description to Verify Domain and Range

The initial description states that the function is translated 4 units to the left and 5 units up. Recall that these translations impact the graph horizontally and vertically but do not affect the set of x-values (domain). The vertical translation shifts the lowest point of the graph (minimum value of y) from 0 to 0+5=5, affecting the range.

Key concepts

Horizontal Translation

Horizontal translations shift the graph of a function left or right along the x-axis. This transformation changes the input value, affecting the function's argument inside the parentheses.
For horizontal shifts:

• Move right: Replace every occurrence of x with \( x-h \) where h is the number of units to shift right.
• Move left: Replace x with \( x+h \) where h is the number of units to shift left.

In this exercise, we start with the function \( y=x^2 \). To translate it 4 units to the left, we replace x with \( x+4 \):
\[ y = (x+4)^2 \]
This operation shifts every x-value on the graph 4 units to the left.

Vertical Translation

Vertical translations shift the graph of a function up or down along the y-axis. This transformation affects the output value of the function.
For vertical shifts:

• Move up: Add a constant \( k \) to the entire function, where k is the number of units to shift up.
• Move down: Subtract a constant \( k \) from the entire function, where k is the number of units to shift down.

In the exercise, after applying the horizontal translation, the function is \( y=(x+4)^2 \). To translate it 5 units up, we add 5 to the entire function:
\[ y = (x+4)^2 + 5 \]
This operation shifts the entire graph 5 units upward.

Domain and Range

Understanding the domain and range of a function is crucial for recognizing which values are valid inputs (domain) and which values can be outputs (range).
The domain of a function represents all possible x-values. For the basic quadratic function \( y = x^2 \), the domain is all real numbers: \[ (-\infty, \infty) \]
Translations do not affect the domain, so even after transforming the function to \( y=g(x)=(x+4)^2 + 5 \), the domain remains:
\[ (-\infty, \infty) \]
The range of a function represents all possible y-values. For \( y = x^2 \), the range is \[ [0, \infty) \] as the square of any real number is non-negative.
After translating the function 5 units up, every point on the graph is shifted upward by 5 units. Therefore, the lowest y-value moves from 0 to 5, making the range:
\[ [5, \infty) \]
These transformations help us verify that the domain and range of the translated function are consistent with the described translations.