Problem 10 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. The graph of \(f(x)=(x-4)^{2}\) is the graph of \(y=x^{2}\) shifted __________ 4 units.

Step-by-Step Solution

Answer

The graph of \(f(x) = (x - 4)^2\) is the graph of \(y = x^2\) shifted to the right by 4 units.

Step 1: Recognize the Shifted Function

The given function is \(f(x) = (x - 4)^2\). This can be seen as a transformed version of the function \(f(x) = x^2\). The transformation has been done by replacing \(x\) in the function \(f(x)\) with \((x - 4)\).

Step 2: Interpret the Shift

Such a transformation represents a horizontal shift in the original graph \(f(x) = x^2\). The value after the minus sign in \((x - 4)^2\) represents the magnitude of horizontal shift. The minus sign indicates that the shift is towards the right side of the graph. Therefore, the function \(f(x) = (x - 4)^2\) represents a shift of the graph \(f(x) = x^2\) 4 units to the right.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with its highest degree being two. This means it involves a squared variable, usually represented as \(y = ax^2 + bx + c\). The graph of a quadratic function forms a curve called a parabola.
The simplest form of a quadratic function is \(f(x) = x^2\), where:

• The graph of \(f(x) = x^2\) is a parabola that opens upward.
• The vertex (the lowest or highest point, depending on the orientation) of this parabola is at the origin \((0,0)\) on the coordinate plane.
• The axis of symmetry for this parabola is the vertical line that passes through the vertex, in this case, the y-axis \(x = 0\).
• Quadratic functions are used to model various real-world scenarios, notably those involving projectiles due to their trajectory-path resemblance.

Horizontal Shift

A horizontal shift is a type of transformation that moves a graph left or right from its original position. In the context of quadratic functions, the standard form \(f(x) = (x-h)^2\) incorporates horizontal shifts easily.
When you see \(f(x) = (x - 4)^2\), it means:

• The graph of the original quadratic function \(f(x) = x^2\) is being shifted.
• The parameter \((x - 4)\) inside the function indicates the shift.
• Minus 4 signifies that each point of the graph moves 4 units to the right.

In general, if \(h\) is positive in \((x - h)\), the graph shifts to the right by \(h\) units. If it’s negative, it shifts to the left.

Graph Transformation

Graph transformation refers to various ways we can modify the picture of a graph on a coordinate plane. This includes translating (shifting), reflecting, scaling, or rotating the original graph. When talking about quadratic functions like \(f(x) = x^2\), transformations are more straightforward but visible exams on how the shape and position of the graph change.
For the function \(f(x) = (x - 4)^2\):

• **Translation:** The graph moves horizontally from its original spot, specifically along the x-axis.
• **Shape and Size:** The parabolic shape remains the same, maintaining its upward-curved form.
• **Vertex Movement:** The vertex of \(f(x) = x^2\), which was originally at \((0,0)\), relocates to \((4,0)\) after the shift due to transformation.

Understanding graph transformations helps in analyzing how changes in function equations affect their graphs in predictable ways.