The graph of `g(x)=(3x-9)(x-1)` is a parabola in the xy-plane. If the vertex of this parabola has coordinates (h, k), what is the value of `h-k`?

Answer

To solve the problem, we need to find the vertex of the parabola given by the equation \( g(x) = (3x - 9)(x - 1) \) and then calculate the value of \( h - k \), where \( (h, k) \) are the coordinates of the vertex.

### Step 1: Expand the equation
First, we need to expand the equation \( g(x) = (3x - 9)(x - 1) \).

\[ g(x) = 3x^2 - 3x - 9x + 9 = 3x^2 - 12x + 9 \]

### Step 2: Identify coefficients
The quadratic equation is now in the standard form \( g(x) = ax^2 + bx + c \), where:
- \( a = 3 \)
- \( b = -12 \)
- \( c = 9 \)

### Step 3: Find the x-coordinate of the vertex
The x-coordinate of the vertex \( h \) can be found using the formula:

\[ h = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \):

\[ h = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2 \]

### Step 4: Find the y-coordinate of the vertex
To find the y-coordinate \( k \), we substitute \( x = h = 2 \) back into the equation \( g(x) \):

\[ g(2) = 3(2)^2 - 12(2) + 9 \]

Calculating this:

\[ g(2) = 3(4) - 24 + 9 = 12 - 24 + 9 = -3 \]

Thus, \( k = -3 \).

### Step 5: Calculate \( h - k \)
Now we can find \( h - k \):

\[ h - k = 2 - (-3) = 2 + 3 = 5 \]

### Final Answer
The value of \( h - k \) is \( \boxed{5} \).

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