Problem 27 - Algebra and Trigonometry

Question

\(27-32\) : 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. \(f(x)=x^{2},\) shift upward 3 units and shift 2 units to the right

Step-by-Step Solution

Answer

The transformed function is \( f(x) = (x-2)^2 + 3 \).

Step 1: Initial Function

We start with the original function given in the problem, which is a quadratic function: \( f(x) = x^2 \).

Step 2: Vertical Shift Upward

To shift the graph of the function upward by 3 units, we add 3 to the original function. This means our new function becomes: \( f(x) = x^2 + 3 \).

Step 3: Horizontal Shift to the Right

To shift the graph 2 units to the right, we replace \(x\) with \((x-2)\) in the function. The transformation results in: \( f(x) = (x-2)^2 + 3 \).

Key Concepts

Vertical Shift

A vertical shift in a function refers to moving the entire graph up or down along the y-axis. It does not change the shape of the graph, only its position. To perform a vertical shift:

• Add a constant to the function if you want to move the graph upward.
• Subtract a constant from the function to move the graph downward.

For example, if we have a function like \( f(x) = x^2 \), applying a vertical shift upward by 3 units would result in the function \( f(x) = x^2 + 3 \).
In this transformation, every point on the original graph \( y = x^2 \) would move up by 3, causing the entire parabola to rise but maintain its characteristic 'U' shape. This is because the added constant shifts the y-values directly upwards, without affecting the x-values.
Vertical shifts are simple adjustments that can alter the position of the graph without impacting its basic characteristics.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right on the x-axis. Unlike vertical shifts, horizontal shifts affect the input, or \( x \), of the function. To conduct a horizontal shift:

• Replace \( x \) with \( x - k \) in the function to shift the graph to the right by \( k \) units.
• Replace \( x \) with \( x + k \) to shift the graph to the left by \( k \) units.

This concept can initially be tricky because the direction of the shift may seem counterintuitive. A shift to the right, for instance, involves a negative sign: \( (x - 2) \) moves the graph 2 units to the right.
For our quadratic function \( f(x) = x^2 \), if we replace \( x \) with \( x-2 \), the graph is effectively moved 2 units to the right, resulting in the transformed function \( f(x) = (x-2)^2 \).
These shifts change where the shape appears on the graph but do not change its general appearance.

Quadratic Function

Quadratic functions are a key concept in algebra, represented by \( f(x) = ax^2 + bx + c \). This type of function creates a graph called a parabola, a symmetrical, U-shaped curve. The simplest quadratic function is \( f(x) = x^2 \), having its vertex at the origin (0,0) and opening upwards.
Quadratic functions display certain features that are easy to identify:

• The vertex: the highest or lowest point on the graph, depending on whether it faces up or down.
• The axis of symmetry: a vertical line through the vertex, dividing the parabola into two mirror-image halves.
• The direction: determined by the coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Transformations such as vertical and horizontal shifts modify the position of the quadratic graph, allowing it to reflect various real-world scenarios while maintaining its fundamental shape. As we explored earlier, combining a vertical shift of +3 and a horizontal shift to the right by 2 transforms \( f(x) = x^2 \) into \( f(x) = (x-2)^2 + 3 \), shifting the vertex from (0,0) to (2,3). Understanding quadratic functions and their transformations can provide deep insights into both algebraic operations and their graphical interpretations.