Problem 23 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{array}{l}f(x)=x^{2} \\\g(x)=(x+2)^{2}\end{array}$$

Step-by-Step Solution

Answer

First, sketch the parent function \(f(x) = x^2\), which is a parabola with vertex at (0, 0) and opening upwards. Next, sketch the graph of \(g(x) = (x+2)^2\) by shifting the graph of f(x) 2 units to the left. The graph of g(x) is also a parabola with vertex at (-2, 0) and opening upwards. Finally, label both graphs to differentiate between f(x) and g(x).

Step 1: Identify the parent function

The parent function for both f(x) and g(x) here is a quadratic function, given by \(f(x)=x^2\).

Step 2: Identify the transformation involved

The transformation between f(x) and g(x) is a horizontal shift. g(x) is formed by shifting the graph f(x) 2 units to the left. This can be seen in the expression for g(x): \(g(x)=(x+2)^2\).

Step 3: Sketch the graph of the parent function f(x)

First, sketch the graph of the parent function: \(f(x)=x^2\). A quadratic function has a parabola shape, and the graph of \(f(x)=x^2\) has its vertex at (0, 0) and is symmetric about the y-axis. Since the coefficient of the x^2 term is positive, the parabola opens upwards.

Step 4: Apply the transformation to the parent function

To sketch the graph of g(x), apply the transformation to the parent function f(x). In this case, the transformation is a horizontal shift 2 units to the left. The vertex of the graph of g(x) will now be at (-2, 0), and the parabola will have the same shape as the graph of f(x).

Step 5: Sketch the graph of the transformed function g(x)

Using the information from Step 4, draw the graph of \(g(x)=(x+2)^2\) on the same set of axes as the graph for f(x). Make sure to maintain the same shape as f(x), but with the vertex shifted to (-2, 0) and still opening upwards.

Step 6: Label the graphs

To finish, make sure to label both graphs appropriately so that it is clear which graph represents f(x) and which represents g(x).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is characterized by its highest exponent of the variable being 2. This makes the general form of a quadratic function look like this: \( f(x) = ax^2 + bx + c \).Where:

• \(a\), \(b\), and \(c\) are constants.
• \(a\) is not zero, as this would make it a linear equation.

For the specific function \(f(x) = x^2\), the constants \(b\) and \(c\) are zero. This simplifies it to the basic form of a quadratic function.These graphs take the shape of a 'U', known as a parabola. The parabola for \(f(x) = x^2\) is symmetric about the y-axis and its vertex, or lowest point, is at the origin (0,0). The graph opens upwards because the coefficient of \(x^2\) is positive.

Horizontal Shift

In graph transformations, a horizontal shift is a type of translation that moves the graph of a function left or right along the x-axis.This occurs without affecting the shape of the graph.It's represented in the function by adding or subtracting a value inside the function's argument.For example, in the transformation from \(f(x) = x^2\) to \(g(x) = (x+2)^2\), there is a horizontal shift. The '+2' inside the parentheses indicates a shift 2 units to the left.Here's how to determine direction:

• If the expression inside the parentheses is \(x-k\), the graph shifts \(k\) units right.
• If the expression is \(x+k\), the graph shifts \(k\) units left.

In this exercise, the graph of \(f(x)\) shifted left, so the vertex changes from (0,0) to (-2,0), keeping the parabola's shape intact.

Parent Function

The parent function is the simplest form of a function type, which serves as the building block for more complex functions.For quadratic functions, the parent function is \(f(x) = x^2\).This particular function is often used as the base graph.Transformations such as shifting, stretching, or reflecting are applied to the parent function to develop new functions.When working with transformation problems, like in this exercise, identifying the parent function is usually the first step.Understanding it allows you to better visualize how each transformation modifies the graph.

Vertex of a Parabola

The vertex of a parabola is a critical point that represents the peak or the lowest point of the graph, depending on its orientation.For a parabola that opens upward, like \(f(x) = x^2\), it's the lowest point.The vertex offers valuable information:

• Its x-coordinate shows where the graph achieves its minimum or maximum value.
• The y-coordinate of the vertex is the value of the function at that point.

In the parent function \(f(x) = x^2\), the vertex is at (0,0).With a horizontal shift, the exercise showed that the function \(g(x) = (x+2)^2\) moves the vertex to (-2,0), just shifting it along the x-axis while retaining the same y-value.