Problem 15 - Beginning and Intermediate Algebra

Question

Given the following pairs of functions, explain how the graph of \(g(x)\) can be obtained from the graph of \(f(x)\) using the transformation techniques. $$f(x)=x^{2}, g(x)=(x+2)^{2}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = (x+2)^2\) can be obtained from the graph of \(f(x) = x^2\) by applying a horizontal shift of 2 units to the left. This results in a new parabola with its vertex at (-2, 0) and still opening upwards.

Step 1: Identify the transformation from \(f(x)\) to \(g(x)\)

To identify the transformation, we need to compare the two functions and see what changes we need to make to \(f(x)\) so that it becomes \(g(x)\). We can see that \(f(x) = x^2\) and \(g(x) = (x+2)^2\). So, the only difference between the functions is the addition of "+2" inside the exponent. This means we have a horizontal shift of 2 units to the left.

Step 2: Visualize the transformation on a graph

On a coordinate plane, plot the graph of \(f(x) = x^2\). This is a parabola with its vertex at the origin (0,0) and opening upwards. Now, to obtain the graph of \(g(x) = (x+2)^2\), we need to apply the horizontal shift of 2 units to the left. To do this, move every point on the graph of \(f(x)\) 2 units to the left.

Step 3: Describe the resulting graph of \(g(x)\)

After applying the horizontal shift of 2 units to the left, the graph of \(g(x) = (x+2)^2\) is a new parabola with its vertex at (-2, 0) and still opens upwards. This graph is obtained from the graph of \(f(x) = x^2\) by moving each point on the graph of \(f(x)\) 2 units to the left.

Key Concepts

Graph Shifts

When we talk about graph shifts, we're diving into how we can move a graph around the coordinate plane. This involves changing the position of a graph without altering its shape.
For any function, we can shift its graph along the x-axis, the y-axis, or both. These can be either horizontal or vertical shifts. For example, adding or subtracting a number inside the function's argument can lead to a horizontal shift, while adding or subtracting outside the function can lead to a vertical shift.
These shifts help us easily visualize changes in the function's behavior, making them an integral part of understanding function transformations. Graph shifts are particularly useful for understanding transformations of common functions like parabolas, lines, and other curves.

Parabola Transformation

Parabolas are U-shaped curves that come from quadratic functions like the standard form, \(f(x) = x^2\). Parabola transformations involve changing the position and shape of a parabola through different modifications.
These transformations include shifting the graph horizontally or vertically, stretching or compressing it, and reflecting it across axes. Each type of transformation has a specific effect:

• Horizontal shifts involve adding or subtracting a constant to the variable \(x\) within the function.
• Vertical shifts involve adding or subtracting a constant to the entire function itself.
• Reflecting depends on multiplying the function by -1.

Understanding these transformations allows for a more straightforward approach to manipulating and graphing quadratic functions.

Horizontal Shift

A horizontal shift occurs when the graph of a function is moved left or right along the x-axis.
When a function appears as \(g(x) = f(x + c)\), the graph shifts \(c\) units to the left if \(c\) is positive, and \(c\) units to the right if \(c\) is negative.
In our exercise, comparing \(f(x) = x^2\) and \(g(x) = (x+2)^2\) reveals a horizontal shift of 2 units to the left. This shift means that each point on the graph of \(f(x)\) moves 2 units leftward to form the graph of \(g(x)\). This is a fundamental transformation that is often one of the first steps in understanding function transformations.

Vertex Form of a Parabola

The vertex form of a parabola is a convenient way to express quadratic functions, making it easier to identify the transformations applied to a parabola. This form is written as: \[ y = a(x-h)^2 + k \] where \(h, k\) are the coordinates of the vertex of the parabola.
This form directly shows the horizontal and vertical shifts experienced by the parabola.
In this equation:

• \(h\) indicates a horizontal shift: if \(h\) is positive, the shift is to the right, and if negative, to the left.
• \(k\) indicates a vertical shift: if \(k\) is positive, the shift is upwards, and if negative, downwards.

By writing a parabola in vertex form, identifying and applying transformations such as shifts or reflections becomes much more manageable, allowing us to effortlessly graph quadratic functions with precision.