Let's solve the problem step by step:

Part (a) - Write $f(x)$ in the form $(x + p)^2 + q$

We are given the quadratic function:

$f(x) = x^2 + 4kx + (3 + 11k)$

We need to complete the square to express this function in the form $(x + p)^2 + q$.

Step 1: Group the quadratic and linear terms

$f(x) = (x^2 + 4kx) + (3 + 11k)$

Step 2: Complete the square

To complete the square, we need to take half of the coefficient of $x$ (which is $4k$), square it, and add/subtract it inside the function.

Half of $4k$ is $2k$, and squaring it gives $(2k)^2 = 4k^2$.

Thus: $f(x) = (x^2 + 4kx + 4k^2) - 4k^2 + (3 + 11k)$

$f(x) = (x + 2k)^2 - 4k^2 + 3 + 11k$

Step 3: Simplify the constant term

Simplifying the constants: $f(x) = (x + 2k)^2 + (-4k^2 + 11k + 3)$

Thus, the quadratic in completed square form is: $f(x) = (x + 2k)^2 + (11k - 4k^2 + 3)$

Here, $p = 2k$ and $q = 11k - 4k^2 + 3$.

Part (b) - Sketch the graph for $k = 1$

Now, we substitute $k = 1$ into the function and sketch it.

Step 1: Substitute $k = 1$ into the completed square form

$f(x) = (x + 2(1))^2 + (11(1) - 4(1)^2 + 3)$ $f(x) = (x + 2)^2 + (11 - 4 + 3)$ $f(x) = (x + 2)^2 + 10$

So the function is $f(x) = (x + 2)^2 + 10$.

Step 2: Analyze the graph

• Vertex (minimum point): The vertex occurs at $x = -2$ (from $x + 2 = 0$) and $y = 10$. Therefore, the vertex is $(-2, 10)$.
• Y-intercept: To find the y-intercept, set $x = 0$: $f(0) = (0 + 2)^2 + 10 = 4 + 10 = 14$ So, the y-intercept is $(0, 14)$.
• X-intercepts: Since the minimum value of $f(x) = (x + 2)^2 + 10$ is 10 (which is above the x-axis), there are no x-intercepts.

The graph is a parabola opening upwards with vertex $(-2, 10)$ and y-intercept at $(0, 14)$. There are no x-intercepts.

Would you like me to provide a graph visualization of this function?

Here are five follow-up questions related to this problem:

1. How can we identify the vertex of a quadratic function from its completed square form?
2. What is the geometric significance of completing the square?
3. How do the values of $k$ affect the shape and position of the parabola?
4. How can we derive the x-intercepts of a quadratic function?
5. Why does the parabola not have x-intercepts when $k = 1$?

Tip:

When completing the square, always remember to carefully handle the constant terms to avoid errors in simplifying the equation.