91Ó°ÊÓ

A quadratic function is a type of polynomial function where the highest-degree term is squared. The general form is given by:
\textbf{\[ y = ax^2 + bx + c \]}
Here:

• \textbf{a}, \textbf{b}, and \textbf{c} are constants.
• \textbf{a} determines the width and direction of the parabola.
• \textbf{b} and \textbf{c} affect the position of the graph.

When you plot a quadratic function, it creates a U-shaped graph called a \textbf{parabola}.
The simplest quadratic function is \textbf{\( y = x^2 \)}, which has its vertex at (0,0) and opens upwards.

Graph transformations involve shifting or stretching the graph of a function in different ways. There are several types of transformations including:

• **Horizontal Shifts**: Moves the graph left or right.
• **Vertical Shifts**: Moves the graph up or down.
• **Reflections**: Flips the graph over a line, like the x-axis or y-axis.
• **Stretching/Compressing**: Changes the width or height of the graph.

In this exercise, we focus on horizontal shifts.
For instance, translating the graph of \textbf{\( y = x^2 \)} by \textbf{h} units horizontally results in a new function:

**\( y = (x - h)^2 \)**
If \textbf{h} is positive, the graph shifts to the right.
If \textbf{h} is negative, the graph shifts to the left.
This transformation formula helps us determine what shift was applied to make the graph pass through a specific point.
In our example, we saw that \textbf{h = 1} because using \textbf{(5, 16)} made sense when substituted back into the equation.

The vertex form of a quadratic function is an alternative way to express the equation of a parabola. It highlights the vertex (the highest or lowest point) of the parabola and is written as:
**\( y = a(x - h)^2 + k \)**
Here:

• \textbf{(h, k)} is the vertex of the parabola.
• \textbf{a} determines the width and direction of the parabola.

This form is useful for easily identifying vertex position, making it simpler to perform translations.
In the given exercise, converting the standard form \textbf{\( y = x^2 \)} to the vertex form \textbf{\( (x - h)^2 \)} allows us to clearly see the effect of the horizontal translation.
To find the proper translation, we substituted the point \textbf{(5,16)} and solved for \textbf{h}. We discovered that the graph passes through \textbf{ (5,16)} if \textbf{h = 1}, translating the vertex from \textbf{(0,0)} to \textbf{(1,0)}.