Problem 16 - Beginning and Intermediate Algebra

Question

Given the following pairs of functions, explain how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) using the transformation techniques. $$f(x)=x^{2}, g(x)=(x-3)^{2}$$

Step-by-Step Solution

Answer

To obtain the graph of \(g(x)=(x-3)^2\) from the graph of \(f(x)=x^2\), apply a horizontal shift of the graph of \(f(x)\) 3 units to the right. This transformation moves the vertex of the parabola from \((0,0)\) to \((3,0)\), while maintaining its upward-facing orientation.

Step 1: Identify base function and transformation function#

In this exercise, the base function (parent function) is \(f(x)=x^2\), and we want to find the transformation that will make it into the function \(g(x)=(x-3)^2\). The given function \(g(x)\) represents a transformed function of \(f(x)\). So, we will analyze how this transformation occurs in terms of shifting the graph of \(f(x)\).

Step 2: Understand the shift of the graph#

The function \(g(x)=(x-3)^2\) represents a horizontal shift of the graph of the base function \(f(x)=x^2\). In particular, this shift occurs in the following way: - Replace x with (x-3) in \(f(x)\) to get \(g(x)\).

Step 3: Identify the horizontal shift#

The function \(g(x)=(x-3)^2\) represents a horizontal shift of the graph of \(f(x)=x^2\) to the right. The shift is determined by the numerical value inside the parentheses, which is 3 in this case. To see why this is a shift to the right, consider that when \(x=0\), the function evaluates to: $$g(0) = (0-3)^2 = 3^2 = 9$$ So, when x is 0, the value of g(x) is equal to the value of f(x) at x = 3. This indicates that the function's graph has shifted 3 units to the right, as the vertex of the parabola has moved from \((0,0)\) to \((3,0)\).

Step 4: Describing the transformation process#

Now that we have identified the essential transformation between the two functions, we will describe the process: 1. Start with the graph of the base function \(f(x)=x^2\), which is a parabola with its vertex at the origin and facing upward. 2. Apply the horizontal shift to the graph of \(f(x)\) by replacing x with (x-3). This results in the function \(g(x)=(x-3)^2\). 3. Move the graph of \(f(x)\) 3 units to the right. The vertex of the parabola will now be at the point \((3,0)\), and the graph will still be facing upward. By going through these steps, we have transformed the graph of the base function \(f(x)=x^2\) into the graph of the function \(g(x)=(x-3)^2\).

Key Concepts

Horizontal Shift

A horizontal shift moves a graph left or right in the coordinate plane. For instance, in the function transformation from \(f(x)=x^2\) to \(g(x)=(x-3)^2\), a horizontal shift occurs. Here’s how it works:

• The expression \((x-3)\) indicates a shift to the right by 3 units.
• The value '3' is the amount of movement along the x-axis.

To understand this intuitively, if you replace \(x\) with \(x-3\), you're decreasing x-values by 3 to keep the expression balanced. Thus, each point on the graph of \(f(x)\) is relocated 3 units rightward, creating \(g(x)\). This kind of horizontal adjustment is common in graph transformations and is often straightforward once you spot the pattern.

Parabola

A parabola is a symmetrical, U-shaped curve that arises when graphing a quadratic function. For the function \(f(x)=x^2\), you have the most basic form of a parabola that sets a foundation for learning graph transformations.

• Parabolas have a distinctive shape, opening upward or downward.
• The standard form, called the parent parabola, is \(f(x)=x^2\).

Understanding what constitutes a parabola helps in grasping graph manipulations. When we transform \(f(x)\) into \(g(x)=(x-3)^2\), the fundamental shape of the parabola remains consistent. It still looks like a U-shape that opens upwards, but its position in the coordinate grid has changed.

Vertex

The vertex of a parabola is the point where it changes direction, effectively acting as its peak or trough. For \(f(x)=x^2\), the vertex sits at the origin, \((0,0)\).In transformations, knowing the vertex’s role is crucial. Here’s what happens when you shift from \(f(x)\) to \(g(x)=(x-3)^2\):

• The vertex moves horizontally 3 units to the right, landing at \((3,0)\).
• The vertex's new position delineates the new "center" of the graph.

Recognizing how transformations affect the vertex helps make sense of shifts and other changes—the vertex essentially navigates the transformation.

Graph Transformations

Graph transformations empower you to modify the visual representation of functions by altering certain elements in their expressions. They include translations, reflections, stretchings, and compressions.
When observing how \(f(x)=x^2\) transforms into \(g(x)=(x-3)^2\), you specifically deal with a translation:

• This involves moving the entire graph without altering its shape.
• The transformation is purely horizontal in this instance—moving rightward.

Understanding graph transformations makes algebra more visual. Instead of seeing functions solely as equations, transformations let you view them as dynamic structures you can adjust, modify, and manipulate over the Cartesian plane. This makes it easier to predict how similar changes alter other functions.