Problem 37 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=\frac{1}{4}(x+1)^{2}\)

Step-by-Step Solution

Answer

The graph is based on \(y = x^2\), shifted 1 unit left, and compressed vertically by a factor of 4.

Step 1: Identify the base function

Identify the base function from which the transformation begins. The given function is based on the quadratic function \(y = x^2\).

Step 2: Horizontal Shift

The function \(f(x)=(x + 1)^2\) indicates a horizontal shift. The term \((x + 1)\) shifts the graph of \(y = x^2\) 1 unit to the left.

Step 3: Vertical Compression

Apply vertical compression to the function. The \((x + 1)^2\) is multiplied by \(\frac{1}{4}\), which compresses the graph vertically by a factor of 4.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental type of mathematical function characterized by an equation in the form of \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a curve called a **parabola**. Parabolas have distinct features like the vertex, axis of symmetry, and direction of opening, which can be upwards or downwards depending on the sign of the coefficient \( a \).
Quadratic functions are pivotal in algebra and calculus due to their applications in physics, engineering, and many other fields. They are also the starting point for exploring more complex transformations of functions.
In the given exercise, the base function is \( y = x^2 \), which is the simplest form of a quadratic function, highlighting its symmetrical and unimodal nature around the vertical axis that passes through its vertex.

Horizontal Shift

A horizontal shift in a function's graph is a transformation that moves the entire graph left or right along the horizontal axis. It doesn't alter the shape of the graph, only its position. For the function \( f(x) = \frac{1}{4}(x+1)^2 \), the sub-expression \((x+1)\) indicates a horizontal shift.
This specific transformation means that we need to move the graph of the base function \( y = x^2 \) 1 unit to the left. This is because replacing \( x \) with \( x+1 \) in the function suggests shifting each point on the curve in the negative \( x \) direction. Thus, any point \( (x, y) \) on the original parabola becomes \( (x-1, y) \) on the transformed graph.
Horizontal shifts are crucial in aligning graphs to desired positions on the axis and serve as a tool for manipulating the placement of function graphs horizontally without changing their overall structure.

Vertical Compression

Vertical compression is a transformation that alters the vertical scale of a graph by reducing the distance of the points from the horizontal axis. Unlike horizontal shifts, vertical compressions change the shape of the graph by making it "shorter" vertically, though it remains symmetrical.
In the function \( f(x) = \frac{1}{4}(x+1)^2 \), the coefficient \( \frac{1}{4} \) inside the equation implies a vertical compression. To comprehend this, imagine multiplying every \( y \)-coordinate on the graph of \( (x+1)^2 \) by \( \frac{1}{4} \). Points that were once higher up on the \( y \)-axis are now closer to the x-axis, compressing the graph vertically by a factor of 4.
Vertical compression ensures graphs retain their horizontal structure but with a squashed appearance along the \( y \)-direction, thereby maintaining their overall symmetry and general pattern.