91Ó°ÊÓ

When we talk about a parabola in mathematics, we're referring to the graph of a quadratic function. The simplest form of a quadratic function is given by the equation: \[ y = x^2 \].This equation represents a parabola that opens upwards and has its vertex at the origin (0,0). The vertex is the highest or lowest point on the graph of the parabola.

Key characteristics of a parabola include:

• The vertex is where the curve changes direction.
• It is symmetrical about a vertical line through its vertex, known as the axis of symmetry.
• The parabola extends infinitely in both directions along the x-axis.

In transforming the graph of a parabola, such as translating it upwards or downwards or stretching it horizontally, the base \( y = x^2 \) acts as a starting point for determining how these operations change the graph.

A horizontal stretch occurs when the graph of a function is widened along the x-axis. This means the points on the graph move further apart horizontally.

Consider the graph transformation discussed in the exercise, given as: \[ y_2 = \left( \frac{1}{2} x \right)^2 - 1 \].

The factor by which the function is stretched horizontally is determined by the reciprocal of the coefficient of x inside the function. In this case, the reciprocal of \( \frac{1}{2} \) is 2, indicating a horizontal stretch by a factor of 2.

• This means each x-coordinate on the initial parabola \( y = x^2 \) is multiplied by 2.
• The result is a wider parabola that maintains its basic shape but covers more horizontal space.
• Despite stretching, the vertex of the parabola is not affected vertically by the horizontal stretch alone.

In contrast to horizontal stretch, a horizontal compression squeezes the graph towards the y-axis. This makes the graph appear narrower than the original. If you've ever zoomed out on a picture, you might have seen a similar effect.

The exercise specifies the function: \[ y_3 = (2x)^2 - 1 \].

The graph is horizontally compressed by the factor given by the reciprocal of the 2 inside the function, which is \( \frac{1}{2} \). This indicates a horizontal compression by a factor of \( \frac{1}{2} \).

• Each x-coordinate of the points on the base graph \( y = x^2 \) is half of its original.
• This results in a narrower graph, resulting in the parabola looking "steeper".
• The vertex remains at the same y-level after this transformation.

A translation of a graph involves shifting it in one or more directions on the Cartesian plane. The graph maintains its original shape and orientation but moves to a different location.

The exercises with functions \( y_1 \), \( y_2 \), and \( y_3 \) all involve a downward translation by 1 unit. This means all the points on these parabolas move one unit down on the y-axis.

Translations can be identified by looking at the constants added or subtracted from the function. For example,

• In \( y_1 = x^2 - 1 \), the \(-1\) causes the parabola to shift downward.
• Similarly, \( y_2 \) and \( y_3 \) also have the \(-1\), indicating the same downward translation.

After a vertical translation, the "height" of the vertex changes, specifically moving from (0,0) to (0,-1) for all functions in the exercise. This does not affect the graph's opening direction or width.