To graph the function \( f(x) = -4(x - 4)^2 - 6 \), we first need to identify key features such as the vertex and another point on the parabola.
Step 1: Identify the vertex
The function is in vertex form \( f(x) = a(x - h)^2 + k \).
- Here, \( a = -4 \), \( h = 4 \), and \( k = -6 \).
- Therefore, the vertex is at the point \( (4, -6) \).
Step 2: Plot the vertex
You can plot the vertex on a graph. At this point, the coordinates are:
- \( (4, -6) \)
Step 3: Choose another point on the parabola
To find another point, we can choose a value for \( x \) and compute \( f(x) \). Let's choose \( x = 3 \): \[ f(3) = -4(3 - 4)^2 - 6 \] \[ = -4(-1)^2 - 6 \] \[ = -4(1) - 6 \] \[ = -4 - 6 \] \[ = -10 \] Thus, when \( x = 3 \), \( f(3) = -10 \), giving us the point \( (3, -10) \).
Step 4: Plot the points
Now you have two points to plot:
- The vertex \( (4, -6) \)
- Another point \( (3, -10) \)
Step 5: Sketch the parabola
The parabola opens downward (since \( a \) is negative) and is symmetric around the line \( x = 4 \). You can also find the point to the right of the vertex to make it symmetrical, such as for \( x = 5 \): \[ f(5) = -4(5 - 4)^2 - 6 \] \[ = -4(1)^2 - 6 \] \[ = -4 - 6 \] \[ = -10 \] This gives us another point \( (5, -10) \).
Final points to plot:
- Vertex: \( (4, -6) \)
- Other points: \( (3, -10) \) and \( (5, -10) \)
Graphing:
To visualize the graph:
- Plot the vertex at \( (4, -6) \).
- Plot the points \( (3, -10) \) and \( (5, -10) \).
- Extend the parabola downwards such that it passes through these points and is symmetric about the line \( x = 4 \).
This represents the parabola \( f(x) = -4(x - 4)^2 - 6 \) in the graph from \( x = 0 \) to \( x = 10 \).
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