Question

Describe the transformation of f(x) = x2 represented by g. Then graph each function \(g(x)=(x-4)^2\)

Short answer

The function \(g(x)=(x-4)^2\) is a transformation of the function \(f(x)=x^2\) involving a horizontal shift 4 units to the right. The graph of g(x) is a parabola with a vertex at (4,0), whereas the graph of f(x) is a parabola with a vertex at the origin.

Step-by-step solution

Identify the Transformation

By comparing \(f(x) = x^2\) with \(g(x) = (x-4)^2\), we see that the x in f(x) has been replaced by (x-4) in g(x). This represents a horizontal translation or shift of the graph. Taking the form \(g(x)=(x-h)^2\), h stands for the horizontal shift of the graph. Here, h is 4.

Direction of the Transformation

The sign in front of 4 in \(g(x) = (x-4)^2\) is negative when compared to x, indicating that the graph of function f(x) will be moved 4 units to the right.

Graph the Functions

To graph f(x) and g(x), start by graphing the function \(f(x)=x^2\), which gives a basic parabola with vertex at the origin. Then to graph \(g(x)=(x-4)^2\), take every point on the graph of f(x) and shift it 4 units to the right. This gives the graph of \(g(x)=(x-4)^2\), which is also a parabola but with the vertex at (4,0).

Key concepts

Horizontal Shift

When we talk about the horizontal shift of a graph, we describe the movement of the graph to the left or right along the x-axis.
In the function transformation, this shift occurs when you modify the x-variable inside the function.

• The form of the equation typically looks like: \[ g(x) = f(x-h) \]
• Here, the value of \( h \) controls the direction and magnitude of the shift.
• A positive \( h \) moves the graph to the right, while a negative \( h \) moves it to the left.

Let's decipher this in the given equation \( g(x) = (x-4)^2 \):

• The value \( h = 4 \), so the graph shifts 4 units to the right.
• This transformation affects the entire graph, moving each point the same distance horizontally.

Quadratic Functions

Quadratic functions are forms of polynomial functions, specially recognized by their degree of two.
The basic structure of a quadratic function is: \[ f(x) = ax^2 + bx + c \]

• The most straightforward form is \( f(x) = x^2 \), representing a parabola centered at the origin.
• These functions exhibit symmetry and create a curve known as a parabola.

Quadratics are incredibly useful in various applications due to their predictable structure and properties.

• They are often used in physics, engineering, and various fields of science.
• The symmetry makes them simple and effective in modeling real-world scenarios.

Parabola

A parabola is the curve depicted in the graph of a quadratic function.
It has a distinctive U-shaped form with specific key features:

• Parabolas can open upwards or downwards depending on the sign of the leading coefficient "a" in \( f(x) = ax^2 \).
• In our example, both \( f(x)=x^2 \) and \( g(x)=(x-4)^2 \) open upwards.
• The parabola is always symmetrically centered around its vertical axis.

Understanding parabolas is critical as they have unique properties that can be applied in many areas.

• For instance, the maximum or minimum point of the parabola is significant in optimization problems.
• Parabolas also find use in graphical representations and predictive modeling.

Vertex

The vertex is a critical feature of a parabola, representing its peak or lowest point.
In a quadratic function \( f(x) = ax^2 + bx + c \), you can find the vertex using the formula: \[ x = -\frac{b}{2a} \]

• In a standard form like \( f(x) = x^2 \), the vertex is at the origin (0,0).
• For \( g(x) = (x-4)^2 \), the horizontal shift changes the vertex to (4,0).

Knowing the vertex is beneficial because:

• It helps in graphing the parabola accurately.
• The vertex offers insights into the maximum or minimum value of the quadratic function related to real-world contexts.

Remember, transformations like horizontal shifts mainly change the position of the vertex, which effectively changes the graph's orientation in the plane.