91Ó°ÊÓ

One of the transformations applied to a function is the horizontal shift. This affects the position of the graph along the x-axis. In the given exercise, for the function \( g(x) = 2(x+3)^2 \), we see \((x+3)\).
The \(+3\) indicates a movement to the left. This might seem counterintuitive because there's a plus sign, but it actually moves the graph in the opposite direction along the x-axis.

To understand this, think about solving a simple equation like \( (x + 3) = 0 \). Here, \( x = -3 \). This helps you see why the graph moves three units to the left. This type of shifting is essential when modifying or analyzing functions.

The rule of thumb for horizontal shifts is:

• \(x + n\) means shift left by \(n\) units.
• \(x - n\) means shift right by \(n\) units.

Observing how components inside the square or absolute value affect the graph is a powerful tool in graph transformations.

Vertical scaling transforms the graph of a function by stretching or compressing it along the y-axis. In this exercise, the transformation happens when we multiply the quadratic expression \((x+3)^2\) by 2 to get \( g(x) = 2(x+3)^2 \).
By multiplying by 2, every y-value of the original function \( f(x) = x^2 \) is doubled.

To visualize vertical scaling, imagine stretching a rubber band vertically. This transformation affects the steepness or width of the parabola, making it narrower if the factor is greater than 1, as in this case.

• If the multiplier is greater than 1, the graph stretches (becomes steeper).
• If the multiplier is between 0 and 1, the graph compresses (becomes flatter).

Vertical scaling does not affect the x-values, only how high or low the points on the graph are plotted. This makes it crucial to consider when predicting graph shapes.

A quadratic function is defined as a polynomial of degree 2, often written in the form \( f(x) = ax^2 + bx + c \).
In its simplest form, like \( f(x) = x^2 \), the graph is a U-shaped curve called a parabola.

Quadratic functions are fundamental in mathematics because they describe a variety of real-world phenomena, including projectile motion and area calculations. The vertex of a parabola plays a key role, acting as either a maximum or minimum point of the function depending on the orientation of the parabola.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

In the exercise, the original function \( f(x) = x^2 \) undergoes transformations to produce a new function. These transformations illustrate how modifications to the basic quadratic can result in a wide range of curves useful across different contexts. Understanding the form and shape of these functions is pivotal as they appear frequently in both theoretical and applied mathematics.