Problem 23 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=x^{2}\) by basic transformations: (a) \(y=x^{2}-2\) (b) \(y=(x-1)^{2}+1\) (c) \(y=-2(x+2)^{2}\)

Step-by-Step Solution

Answer

(a) Down 2 units, (b) Right 1 and up 1 unit, (c) Left 2, flip and stretch vertically.

Step 1: Understanding the Parent Function

The parent function here is \( y = x^2 \), which is a standard parabola opening upwards with its vertex at the origin (0, 0). Now we examine how this function can be transformed to each given function.

Step 2: Analyzing Transformation for y = x^2 - 2

The function \( y = x^2 - 2 \) involves a vertical shift. The \(-2\) indicates a downward shift of 2 units from the parent function. The vertex thus moves from (0, 0) to (0, -2) while the shape of the parabola remains unchanged.

Step 3: Analyzing Transformation for y = (x-1)^2 + 1

The function \( y = (x-1)^2 + 1 \) combines a horizontal and vertical shift. The expression \((x-1)^2\) implies a shift to the right by 1 unit. The \(+1\) adds a vertical shift upwards by 1 unit. The vertex moves from (0, 0) to (1, 1), transforming the graph without changing its shape.

Step 4: Analyzing Transformation for y = -2(x+2)^2

The function \( y = -2(x+2)^2 \) includes multiple transformations. The \((x+2)^2\) indicates a shift to the left by 2 units. The \(-2\) outside the squared term reflects the parabola vertically and stretches it by a factor of 2, making it open downwards. The vertex of this transformed parabola is at (-2, 0).

Key Concepts

Parabola Transformation

Understanding how parabolas transform from the parent function, such as \(y = x^2\), requires examining changes in the function's form. Parabolas can be modified using basic transformations, which allow us to shift, stretch, compress, or reflect them. These transformations do not alter the fundamental U-shape of the parabola but rather adjust its position and orientation on the coordinate plane.
Parabola transformation involves several operations, key among them being vertical and horizontal shifts, as well as reflection and stretching. By manipulating the algebraic form of the parent function, we can generate various new parabolic functions. Each transformation step has a specific effect, such as moving the vertex or changing the direction in which the parabola opens.
Let's explore how to achieve these transformations using examples from the original exercise.

Vertical Shift

A vertical shift involves moving a function up or down on the coordinate plane. This transformation modifies the function's output values but does not change the input values.
In the example of \( y = x^2 -2 \), the subtraction of 2 from the \( x^2 \) term represents a vertical shift. Specifically, this downward shift of the parabola occurs by 2 units because we subtract 2 from the \( y \)-values of the parent function \( y = x^2 \).
The new vertex is thus relocated from the origin (0, 0) to (0, -2). The entire parabola moves downward, but the shape remains unchanged, maintaining the standard U-shape of the parabola.

Horizontal Shift

Horizontal shifts change the input values of the function, affecting the \(x\)-coordinates while leaving the \(y\)-coordinates unchanged.
Take the function \( y = (x-1)^2 + 1 \). Here, the expression \((x-1)^2\) shows a horizontal shift. The parabola moves 1 unit to the right because we are essentially adding a 1 to each \(x\)-value required to ``zero'' the expression inside the square, thereby moving the vertex from (0, 0) to (1, 1).
Horizontal shifts help in mapping functions onto different portions of the coordinate plane without distorting the overall shape or orientation of the parabola.

Reflection and Stretching

Reflection flips the parabola over a specific axis, which in the case of a vertical reflection, is over the \(x\)-axis. Stretching or compressing changes the width or steepness of the parabola.
The transformed function \( y = -2(x+2)^2 \) demonstrates both a reflection and vertical stretching. The negative sign in front of the term indicates a vertical reflection, meaning the parabola now opens downwards instead of upwards.
Moreover, the coefficient -2 before the squared term causes vertical stretching, making the parabola steeper compared to the parent function. The "stretch" or "compression" effect is determined by the magnitude; since it is greater than 1, it stretches or elongates the parabola.
Such modifications are crucial for adjusting the appearance of the parabola, controlling both its direction and how "tight" or "wide" it appears.