The graph of the equation above is a parabola in the xy-plane. If `kgt0`, what is the minimum value of y expressed in terms of k?

Options

A. `-7k`
B. `-16k`
C. `-25k`
D. `-73k`

Answer

To find the minimum value of \( y \) expressed in terms of \( k \) for the given parabola \( y = k(x - 1)(x + 9) \) where \( k > 0 \), we can follow these steps:

### Step 1: Expand the equation
First, we need to expand the equation to standard form.

\[ y = k(x - 1)(x + 9) \]

Expanding this, we have:

\[ y = k(x^2 + 9x - x - 9) = k(x^2 + 8x - 9) \]

### Step 2: Rewrite in standard form
Next, we want to rewrite the quadratic in the standard form \( y = a(x - h)^2 + k \) to find the vertex.

\[ y = k(x^2 + 8x - 9) \]

We can complete the square for the quadratic expression \( x^2 + 8x \):

1. Take half of the coefficient of \( x \) (which is 8), square it, and add and subtract it inside the equation.
2. Half of 8 is 4, and squaring it gives \( 16 \).

So, we rewrite:

\[ x^2 + 8x = (x^2 + 8x + 16) - 16 = (x + 4)^2 - 16 \]

Substituting this back into the equation gives:

\[ y = k((x + 4)^2 - 16 - 9) = k((x + 4)^2 - 25) \]

### Step 3: Final form
Now, we can express \( y \) in the standard form:

\[ y = k((x + 4)^2 - 25) = k(x + 4)^2 - 25k \]

### Step 4: Identify the vertex
From the standard form \( y = a(x - h)^2 + k \), we can see that the vertex of the parabola is at \( (-4, -25k) \).

### Step 5: Minimum value of \( y \)
Since \( k > 0 \), the parabola opens upwards, and the minimum value of \( y \) occurs at the vertex. Thus, the minimum value of \( y \) is:

\[ \text{Minimum value of } y = -25k \]

### Final Answer
The minimum value of \( y \) expressed in terms of \( k \) is:

\[ \boxed{-25k} \]