91Ó°ÊÓ

Quadratic functions are a type of polynomial function where the highest degree is 2. These functions take the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a curve called a parabola.
Parabolas have a distinct U-shape, and they can open upwards or downwards depending on the sign of \(a\).

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

Every parabola has a vertex, which is the highest or lowest point of the graph, and an axis of symmetry, which is a vertical line that passes through the vertex and divides the graph into two mirror-image halves.
The vertex form of a quadratic function, \(f(x) = a(x-h)^2 + k\), makes it easier to identify the vertex \((h, k)\).
In the standard quadratic function form \(f(x) = ax^2 + bx + c\), you can find the vertex using the formula \(x = -\frac{b}{2a}\). Understanding quadratic functions is crucial when graphing, as they provide a foundation for identifying key features of more complex functions. These functions are everywhere, from physics to finance, modeling various types of real-world behavior.

Parabola shifts occur when the basic graph of a quadratic function, \(y = x^2\), is moved around the coordinate plane without altering its shape. These shifts include vertical and horizontal translations that affect the graph's position.
A vertical shift occurs when a constant is added or subtracted outside the \(x^2\) term. This constant alters the vertex's \(y\)-coordinate, effectively moving the parabola up or down.
For example, in the function \(f(x) = x^2 + k\):

• If \(k > 0\), the parabola shifts up by \(k\) units.
• If \(k < 0\), the parabola shifts down by\( |k| \) units.

In the initial exercise, \(f(x)=x^{2}+4\) shifts the basic parabola up by 4 units, resulting in a vertex at \((0, 4)\), while \(g(x)=x^{2}-4\) shifts it down by 4 units to \((0, -4)\). Therefore, knowing how these shifts work helps in predicting and graphing the changes not just visually but analytically as well.

The range of a function consists of all possible output values (\(y\)-values) that result from inputting all possible \(x\)-values into the function. For quadratic functions represented by parabolas, finding the range involves understanding the direction in which the parabola opens and its vertex.
For functions like \(f(x) = x^2 + 4\), the vertex is at \((0, 4)\). Since the parabola opens upwards, it means that all \(y\)-values must be larger than or equal to the \(y\)-coordinate of the vertex. Therefore, the range is \(y \geq 4\).
Similarly, for \(g(x) = x^2 - 4\), the vertex is at \((0, -4)\), and again as the parabola opens upwards, the range is \(y \geq -4\).
It's important to note the difference between the range and the domain. While the domain refers to all permissible \(x\)-values, the range focuses strictly on the \(y\)-values. Understanding function ranges is vital, as it allows you to predict bounds and limits of quadratic functions in real-life applications such as sound waves, projectile motions, and economics.