Problem 53 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) stretch vertically by a factor of \(2,\) shift downward 2 units, and shift 3 units to the right

Step-by-Step Solution

Answer

The final equation is \(j(x) = 2(x-3)^2 - 2\).

Step 1: Vertical Stretch

First, we apply the vertical stretch to the function \(f(x) = x^2\). A vertical stretch by a factor of 2 means we multiply the entire function by 2. So, the transformed equation becomes \(g(x) = 2x^2\).

Step 2: Shift Downward

Next, we shift the function downward by 2 units. To do this, we subtract 2 from the function \(g(x) = 2x^2\). This results in the equation \(h(x) = 2x^2 - 2\).

Step 3: Shift to the Right

Finally, we shift the function 3 units to the right. For horizontal shifts, we replace \(x\) with \(x - 3\) in the equation. Thus, the final equation becomes \(j(x) = 2(x-3)^2 - 2\).

Key Concepts

Vertical Stretch

Imagine you have a graph of a quadratic function like \(f(x) = x^2\), a simple parabola that opens upwards. When we talk about a vertical stretch, it's like pulling the graph away from the x-axis, making it "taller" or "narrower." This is achieved by multiplying the function by a factor.
An example is stretching the graph by a factor of 2, transforming it into \(g(x) = 2x^2\). Essentially, this takes every y-value on the graph and doubles it.
Some key points to remember about vertical stretches:

• If the factor is greater than 1, the graph stretches "upwards."
• If the factor is between 0 and 1, it compresses "downwards."

Vertical stretches are one of the simplest ways to modify the appearance of a graph without altering its basic shape.

Horizontal Shift

Horizontal shifts move the graph left or right along the x-axis. This is accomplished by modifying the x-value in the function's equation. In our case, we started with \(h(x) = 2x^2 - 2\) and shifted it to the right by 3 units.
To achieve this, we replace every \(x\) with \(x-3\), transforming our equation to \(j(x) = 2(x-3)^2 - 2\).
Key details about horizontal shifts include:

• Replacing \(x\) with \(x - a\) shifts the graph to the right by \(a\) units.
• Replacing \(x\) with \(x + a\) shifts it to the left by \(a\) units.

Horizontal shifts adjust the graph's position but maintain its overall shape.

Quadratic Function

Quadratic functions are a key type of polynomial function, characterized by their simple yet unique U-shaped graphs also known as parabolas. The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\).
Here, the term \(ax^2\) determines the function's direction and width, while \(b\), and \(c\) affect its position on the graph.
Some essential characteristics of quadratic functions include:

• The vertex, which is the highest or lowest point on the graph.
• The axis of symmetry, a vertical line passing through the vertex.
• The direction of the parabola, opening upwards if \(a > 0\) and downwards if \(a < 0\).

Understanding these characteristics helps us quickly recognize and manipulate quadratic functions.

Graph Transformation

Graph transformation involves modifying the graph's shape or position through various techniques such as stretching, shifting, or flipping it. These transformations are essential tools in graphing and understanding functions.
Our exercise demonstrated a sequence of transformations: starting with a vertical stretch, followed by downward and horizontal shifts.
Transformations apply not only to quadratic functions but to virtually any function. They allow us to manipulate the graph's appearance while preserving its fundamental characteristics.

• Vertical transformations (stretches and shifts) alter the graph's distance from the x-axis.
• Horizontal transformations (shifts and compressions) affect the location relative to the y-axis.
• Combining transformations are a powerful way to analyze and visualize different function behaviors.

Mastery of graph transformations enables us to interpret and predict changes in graphs seamlessly.