Graph the function f(x)=


4(x–4)^2–6.
Plot the vertex. Then plot another point on the parabola. If you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

gragh starts from 0 to 10 from x and y axis

1 answer

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:

  1. The vertex \( (4, -6) \)
  2. 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:

  1. Vertex: \( (4, -6) \)
  2. 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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