91Ó°ÊÓ

Quadratic functions are a special class of functions that produce a parabolic shape on a graph. They are typically expressed in the form \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and the variable \(x\) is squared, giving the equation its distinctive curve. The simplest quadratic function is \(y = x^2\), which yields a parabola that opens upwards, centered on the y-axis with its vertex at the origin (0,0).

Quadratics are important in various fields because they model natural phenomena like projectile motion, where objects follow a parabolic path. Additionally, understanding the graph of a quadratic function helps us solve equations and inequalities involving quadratics, optimize functions, and model real-world situations. The shape of the parabola is defined by its concavity (up or down), width, and the position of its vertex, which can change with transformations.

Transforming a parabola involves shifting, stretching, compressing, or reflecting it across the coordinate axes. These operations are important for analyzing how changes in the equation affect the graph's shape and position.

• **Horizontal Shifts**: In the function \((x + h)^2\), if \(h > 0\), the parabola shifts left by \(h\) units, while if \(h < 0\), it shifts right.
• **Vertical Shifts**: Adding a constant \(k\) moves the graph up or down. For instance, \(y = (x + h)^2 + k\) shifts the parabola \(k\) units vertically.
• **Vertical Stretch/Compression**: Multiplying the function by a constant \(a\) (>1 or between 0 and 1) affects the parabola's width. If \(a > 1\), the graph stretches vertically; if \(0 < a < 1\), it compresses.
• **Reflections**: Multiplying \(x^2\) by a negative flips the parabola over the x-axis, creating a downward open shape.

In our transformation scenario, \(p(x) = \frac{1}{2}(x+4)^2\), the initial parabola \(y = x^2\) is transformed through a 4-unit leftward shift and a vertical compression due to the coefficient \(\frac{1}{2}\). These transformations change the width and the vertex's position of the parabola while maintaining the overall parabolic shape.

The vertex form of a quadratic function is a handy way to express parabolas, emphasizing the vertex's role in the graph's shape and position. It is written as \(y = a(x - h)^2 + k\), where \((h, k)\) represents the vertex of the parabola. This format is useful for easily identifying transformations and locating the vertex.

Understanding vertex form enables us to quickly decipher how a parabola has transformed:

• The expression \((x - h)^2\) indicates a horizontal shift, moving the vertex to \(x = h\).
• The constant \(k\) reflects a vertical shift, lifting or lowering the vertex in the y-direction.
• The coefficient \(a\) influences the parabola's width, indicating whether the graph is vertically stretched (if \(a > 1\)), compressed (if \(0 < a < 1\)), or flipped (if \(a < 0\)).

In the exercise function \(p(x) = \frac{1}{2}(x+4)^2\), the vertex is \(-4, 0\). This vertex form shows a parabolic shift left by 4 units and a compression with a factor of \(\frac{1}{2}\), clearly demonstrating the advantages of using the vertex form to analyze transformations in quadratic functions.