Problem 1 - Algebra 2 and Trigonometry

Question

Let \(\mathrm{f}(x)=x^{2}\) and \(\mathrm{g}(x+2)=(x+2)^{2} .\) Are \(\mathrm{f}\) and \(\mathrm{g}\) the same function? Explain why or why not.

Step-by-Step Solution

Answer

No, \( f(x) \) and \( g(x) \) are not the same because \( g(x) = x^2 + 4x + 4 \), differing from \( f(x) = x^2 \).

Step 1: Understand the Function f(x)

The function \( f(x) = x^2 \) is a simple quadratic function where the output is the square of the input \( x \). This function takes any real number \( x \) and maps it to \( x^2 \).

Step 2: Express Function g(x)

The function \( g(x+2) = (x+2)^2 \) represents a horizontal translation of \( f(x) \). We need to find \( g(x) \) in terms of \( x \). Let \( u = x + 2 \), then \( x = u - 2 \), hence \( g(x) = g(u - 2) = (u)^2 \) where \( u = x + 2 \).

Step 3: Compare the Functions

For \( f(x) \), the output is \( x^2 \). For \( g(x) \), if \( x \) is shifted by 2 (i.e., \( g(x+2) \)), the function becomes \( (x+2)^2 = x^2 + 4x + 4 \). So, \( g(x) = x^2 + 4x + 4 \).

Step 4: Analyze the Difference

In \( f(x) \), every input \( x \) results in \( x^2 \). In \( g(x) \), the additional terms \( 4x + 4 \) appear because of the horizontal shift. This means that \( g(x) \) is not identical to \( f(x) \).

Step 5: Conclusion

Since \( f(x) = x^2 \) and \( g(x) = x^2 + 4x + 4 \) are different expressions, with \( g(x) \) having additional terms compared to \( f(x) \), the functions are not the same.

Key Concepts

Understanding the Quadratic Function

A quadratic function is one of the simplest forms of polynomial functions. The most basic type of quadratic function is given as \( f(x) = x^2 \). In this function, the input \( x \) is squared to get the output. Quadratic functions typically create a U-shaped graph called a parabola.

Key characteristics of a quadratic function include:

• It has one variable raised to the second power, such as \( x^2 \).
• The graph of a basic quadratic function is symmetrical about the y-axis.
• The function has a vertex, which is the lowest or highest point depending on its direction.
• The graph often crosses the x-axis at points called roots or zeros.

For \( f(x) = x^2 \), the parabola is centered at the origin \((0,0)\), opening upwards. This indicates that the values of \( f(x) \) grow larger as \( x \) moves away from zero in either direction.

Function Transformation Insight

Function transformations involve altering the basic form of a function to create a new function. It is like bending, stretching, or shifting an existing graph. Transformations can change the position, size, or orientation of the graph.

Transformations include:

• Vertical Translations: Moving the graph up or down without altering its shape.
• Horizontal Translations: Shifting the graph left or right.
• Reflections: Flipping the graph over a specific axis.
• Stretching and Shrinking: Altering the size of the graph by multiplying or dividing the function.

In the case of the function \( g(x+2) = (x+2)^2 \), we are dealing with a horizontal transformation. Here, the entire graph of the function \( f(x) = x^2 \) is shifted to the left by 2 units.

Exploring Horizontal Translation

A horizontal translation shifts a function left or right on the coordinate plane. When talking about the translation of \( f(x) = x^2 \) to \( g(x+2) = (x+2)^2 \), we can understand it as moving the graph of the parabola 2 units to the left.

Horizontal translations are noted as \( x + h \) in a function's equation. If \( h \) is positive, the shift is to the left. If \( h \) is negative, the shift is to the right. This modifies the input of the function and affects only the position, not the shape or orientation.

• Example: For \( g(x) = (x+2)^2 \), \( h = 2 \) signifies the graph shifts 2 units to the left.
• This does not change the parabolic shape. Instead, it affects where the vertex of the parabola is positioned.
• In this particular case, \( g(x) \) compared with \( f(x) \) leads to different outputs, emphasizing the variables are not equivalent after the shift.

Despite this shift, the general properties, like the U-shape, symmetry, and continuity, remain unchanged.