To graph the function [tex]\( f(x) = -(x+6)^2 \)[/tex], follow these steps:
1. Identify the basic function:
The function [tex]\( f(x) = -(x+6)^2 \)[/tex] is a quadratic function in the form of [tex]\( y = -a(x + b)^2 + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Determine the vertex:
The vertex of a quadratic function [tex]\( y = -a(x + b)^2 + c \)[/tex] is given by the point [tex]\( (h, k) \)[/tex]. Here, the vertex is at [tex]\( (-6, 0) \)[/tex].
3. Determine the shape and direction:
Since the coefficient of the squared term is negative ([tex]\( -1 \)[/tex]), the parabola opens downwards.
4. Plot several points:
Choose several values of [tex]\( x \)[/tex] around the vertex to compute [tex]\( f(x) \)[/tex] and plot these points. For example:
- When [tex]\( x = -10 \)[/tex], [tex]\( f(-10) = -((-10) + 6)^2 = -(4)^2 = -16 \)[/tex]
- When [tex]\( x = -8 \)[/tex], [tex]\( f(-8) = -((-8) + 6)^2 = -(2)^2 = -4 \)[/tex]
- When [tex]\( x = -6 \)[/tex], [tex]\( f(-6) = -((-6) + 6)^2 = -(0)^2 = 0 \)[/tex]
- When [tex]\( x = -4 \)[/tex], [tex]\( f(-4) = -((-4) + 6)^2 = -(2)^2 = -4 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = -((-2) + 6)^2 = -(4)^2 = -16 \)[/tex]
5. Draw the axis:
Draw the x-axis and y-axis on a graph.
6. Plot the points and draw the parabola:
Plot the points calculated above and draw a smooth curve through these points to form the parabola.
Let's visualize these steps without a graphical plot:
- The vertex is at [tex]\( (-6, 0) \)[/tex]
- The parabola opens downward
- Points: [tex]\((-10, -16)\)[/tex], [tex]\((-8, -4)\)[/tex], [tex]\((-6, 0)\)[/tex], [tex]\((-4, -4)\)[/tex], and [tex]\((-2, -16)\)[/tex]
By connecting these points smoothly, you get the graph of the function [tex]\( f(x) = -(x+6)^2 \)[/tex].
