To graph the quadratic function [tex]\( f(x) = -2(x+4)^2 - 3 \)[/tex], we will follow these steps:
1. Identify the Vertex:
The function is in vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex. By comparing [tex]\( f(x) = -2(x+4)^2 - 3 \)[/tex] with the general form:
- We can see that [tex]\( h = -4 \)[/tex] and [tex]\( k = -3 \)[/tex].
Therefore, the vertex of the parabola is at [tex]\( (-4, -3) \)[/tex].
2. Choose a Second Point:
To graph the parabola accurately, we need another point on the graph. Let's choose [tex]\( x = -3 \)[/tex] as our second [tex]\( x \)[/tex]-value.
- Plugging [tex]\( x = -3 \)[/tex] into the function to find the corresponding [tex]\( y \)[/tex]-value:
[tex]\[
f(-3) = -2(-3 + 4)^2 - 3 = -2(1)^2 - 3 = -2 - 3 = -5
\][/tex]
Thus, the second point on the parabola is [tex]\( (-3, -5) \)[/tex].
3. Plot the Points and Draw the Parabola:
- Start by plotting the vertex at [tex]\( (-4, -3) \)[/tex].
- Next, plot the second point at [tex]\( (-3, -5) \)[/tex].
- Draw a smooth curve through these points, making sure it opens downwards (since the coefficient of the squared term, [tex]\(-2\)[/tex], is negative), forming a parabola.
By plotting these points, you can accurately graph the quadratic function [tex]\( f(x) = -2(x+4)^2 - 3 \)[/tex].
