Problem 59 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)-(x-2)^{2} $$

Step-by-Step Solution

Answer

The graph of the given function \(g(x) = (x-2)^{2}\) is a parabola opening upwards having its vertex at (2,0); it is the graph of the standard quadratic function shifted two units to the right.

Step 1: Sketch the standard quadratic function

Begin with a sketch of the standard quadratic function \(f(x)=x^{2}\). This is a parabola that opens upwards and has its vertex at the origin, (0, 0).

Step 2: Identify the transformation

In the function \(g(x) = (x-2)^{2}\), the quadratic term is shifted by 2 units to the right. This indicates a horizontal shift or transformation to the right.

Step 3: Apply the transformation

Apply this shift to every point on the graph of \(f(x)=x^{2}\). This means moving every point on the graph of \(f(x) = x^{2}\) two units to the right. The vertex will now be at (2,0).

Step 4: Sketch the transformed function

Draw the transformed function \(g(x) = (x-2)^{2}\). The graph appears as the graph of \(f(x) = x^{2}\) but shifted two units to the right.

Key Concepts

Graph Transformations

A graph transformation involves altering the graph of a function in a systematic way. In the context of quadratic functions like the one given in the exercise, these changes help shift, stretch, or shrink the graph in various directions.

Transforming graphs is crucial for better understanding how mathematical functions behave. When we say transformation in mathematics, it often relates to some of the following changes:

• Horizontal Shifts: Moving the graph left or right.
• Vertical Shifts: Moving the graph up or down.
• Stretches/Shrinks: Changing the width of the graph.
• Reflections: Flipping the graph across an axis.

The exercise focuses on a horizontal shift. We'll delve into each transformation aspect further. By comprehending these, you can readily predict and sketch transformed functions accurately.

Parabola

The term 'parabola' refers to a specific type of curve, which is the graph of a quadratic function. In its simplest form, a parabola opens upward or downward, symmetrically about its vertex.

Understanding the basic structure of a parabola helps when you perform transformations:

• Vertex: The highest or lowest point of the parabola, depending on its direction.
• Axis of Symmetry: A vertical line that runs through the vertex, splitting the parabola into two mirror-image halves.
• Direction: Parabolas can open upwards or downwards, determined by the sign in front of the quadratic term. For example, \(x^2\) opens upwards.

In the given exercise, the standard quadratic function \(f(x) = x^2\) represents a parabola that opens up with its vertex at the origin. Understanding this form is essential before applying any transformations.

Horizontal Shift

A horizontal shift in a function occurs when you move the entire graph left or right along the x-axis. This type of transformation doesn't alter the shape of the graph, only its position.

In mathematical terms, a horizontal shift involves changing the input, or x-value, of the function:

• For a shift to the right, you subtract from the x-value inside the function: \(f(x) = (x - h)^2\).
• For a shift to the left, you add to the x-value inside the function: \(f(x) = (x + h)^2\).

In the exercise, the function \(g(x) = (x-2)^2\) results in a horizontal shift of the quadratic function two units to the right. This shift alters the position of the vertex from (0,0) to (2,0). While the overall shape and direction of the parabola remain unchanged, its position in relation to the x-axis is different.