91Ó°ÊÓ

A quadratic function is a type of polynomial function of degree 2. This means it has the general form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratic functions often appear in the form \( f(x) = a(x - h)^2 + k \), which is known as vertex form. In this form, it becomes easier to see the graph's vertex and other critical features. The quadratic function \( g(x) = -(x + 4)^2 \) is an example of a function in vertex form, where \( a = -1 \), \( h = -4 \), and \( k = 0 \). This specific form reveals crucial properties such as whether the parabola opens upwards or downwards and the position of the vertex. In our example, since \( a = -1 \), the parabola opens downwards, indicating a concave-down shape.

The vertex of a parabola is a significant point that represents the location where the function reaches its maximum or minimum value. For a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), the vertex is given by the coordinates \( (h, k) \). In this problem, the function \( g(x) = -(x + 4)^2 \) can be rewritten as \( g(x) = -(x - (-4))^2 + 0 \), revealing that the vertex is at \( (-4, 0) \). The vertex helps in sketching the graph since it provides a starting point and is the highest or lowest point on the parabola, depending on whether it opens up or down.

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex, effectively splitting the parabola into two mirror images. For functions in vertex form \( f(x) = a(x - h)^2 + k \), this line is represented by the equation \( x = h \). In our example, with \( h = -4 \), the axis of symmetry is \( x = -4 \). Drawing the axis of symmetry helps in ensuring that the parabola is symmetric and provides a guide for plotting points accurately on both sides of the vertex.

Plotting points is a crucial step when graphing quadratics. Begin with the vertex, which is already known. For \( g(x) = -(x + 4)^2 \), the vertex is \( (-4, 0) \). Next, choose some values for \( x \) to determine other points on the graph. For example, if \( x = -3 \), then \( g(-3) = -(((-3) + 4)^2) = -1 \), so one point is \( (-3, -1) \). Similarly, for \( x = -5 \), \( g(-5) = -(((-5) + 4)^2) = -1 \), giving another point \( (-5, -1) \). Plotting these points and reflecting them across the axis of symmetry helps in sketching an accurate graph. Once enough points are plotted, draw a smooth curve through them, ensuring the parabola reflects its concave-down property.