Question

graph the function $f(x) = 3(x - 6)^2$. plot the vertex. then plot another point on the parabola. if you make a mistake can erase your parabola by selecting the second point and placing it on top of the first.

Explanation:

Step1: Identify the vertex form of a parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For the function \( f(x) = 3(x - 6)^2 \), we can rewrite it as \( f(x) = 3(x - 6)^2 + 0 \). So, \( h = 6 \) and \( k = 0 \), which means the vertex is at \((6, 0)\).

Step2: Find another point on the parabola

Let's choose \( x = 7 \) (we can choose any value for \( x \), \( x = 7 \) is a simple choice). Substitute \( x = 7 \) into the function:
\( f(7) = 3(7 - 6)^2 = 3(1)^2 = 3(1) = 3 \). So, the point \((7, 3)\) is on the parabola. We could also choose \( x = 5 \): \( f(5) = 3(5 - 6)^2 = 3(-1)^2 = 3(1) = 3 \), so \((5, 3)\) is also on the parabola (due to symmetry).

To graph the function:

1. Plot the vertex at \((6, 0)\).
2. Plot another point, for example, \((7, 3)\) (or \((5, 3)\) or any other point found by substituting a value of \( x \) into the function). Then, since the coefficient \( a = 3 \) is positive, the parabola opens upwards, and we can sketch the parabola using the vertex and the other point, and the symmetry of the parabola (it is symmetric about the vertical line \( x = 6 \)).

Answer:

The vertex is at \((6, 0)\). Another point on the parabola is \((7, 3)\) (or \((5, 3)\), etc.). To graph, plot \((6, 0)\) and then a point like \((7, 3)\) and draw the upward - opening parabola symmetric about \( x = 6 \).