Problem 27 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$f(x)=(2 x)^{2}$$

Step-by-Step Solution

Answer

The function \(f(x) = (2x)^2\) is a standard quadratic function \(f(x) = x^2\) which has been stretched by a factor of 2 along the x-axis.

Step 1: Identify the Base Function

The basic function in this case is \(f(x) = x^2\), which is a standard quadratic function.

Step 2: Identify the Transformation

For \(f(x) = (2x)^2\), the base function has been transformed by stretching it by a factor of 2 along the x-axis. So the value of x in the function is doubled before it is squared.

Step 3: Sketch the Graph

Sketch the graph of \(x^2\) and then apply the stretching factor. For \(f(x) = (2x)^2\), for each x-coordinate in the original function, the corresponding y-coordinate will be four times as high. This is because squaring the doubling factor for x results in quadrupling the y-coordinate in the original function.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra. They are functions of the form \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). The graph of a quadratic function is a parabola. It can either open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards like a U, and if \( a \) is negative, it opens downwards like an inverted U. The simplest quadratic function is \( f(x) = x^2 \), which represents a parabola whose vertex is at the origin \((0,0)\).
Key features of quadratic functions include their vertex, axis of symmetry, and intercepts:

• The vertex is the highest or lowest point on the graph.
• The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves, usually through the vertex.
• The intercepts are the points where the graph crosses the axes. For \( f(x) = x^2 \), it crosses the y-axis at \((0,0)\).

Understanding these features helps in graphing and transforming quadratic functions.

Graph Sketching

Graph sketching is a critical skill when working with functions. It involves understanding the shape and characteristics of a graph based on its equation. For a quadratic function like \( f(x) = x^2 \), sketching begins by identifying key points like the vertex and intercepts.
To sketch a quadratic function:

• Plot the vertex: For the basic \( f(x) = x^2 \), the vertex is at (0,0).
• Draw the axis of symmetry: a vertical line passing through the vertex.
• Identify the intercepts: Since \( f(x) = x^2 \) passes through (0,0), this is your intercept.
• Plot additional points: Choose values of \( x \) to find corresponding \( y \)-values, ensuring symmetry around the axis.

While sketching the graph of a function like \( f(x) = (2x)^2 \), remember that transformations will change these points' positions.

Function Transformations

Function transformations alter the appearance and properties of a graph. They include translations, reflections, dilations (stretches/shrinks), and rotations. For quadratic functions, knowing how these transformations change the graph helps in sketching and analyzing them.
For the function \( f(x) = (2x)^2 \), a dilation transformation is applied. This is a horizontal compression or stretching of the graph. Here's what happens in such cases:

• Horizontal stretching/compression: For \( f(x) = (2x)^2 \), each \( x \)-coordinate of the parent function \( f(x) = x^2 \) is multiplied by 1/2 (since \( x \) is replaced by \( 2x \)).
• Vertical stretching: The corresponding \( y \)-values are affected by the squared scaling factor, in this case, 4 (since \( (2x)^2 = 4x^2 \)). Thus, the graph becomes narrower because \( y \) increases faster as \( x \) moves away from zero.

These transformations are visible in the modified graph, affecting how parabolas stretch along axes, scale up or down, and shift in position.