Let's analyze the function $f(x) = x^2 + 4x + 10$ as requested in the image.

a) Rewriting $f(x)$ in vertex form:

To rewrite the quadratic function in vertex form, we complete the square.

1. Start with the given function: $f(x) = x^2 + 4x + 10$

2. Take the coefficient of $x$ (which is 4), divide it by 2, and square it: $\left( \frac{4}{2} \right)^2 = 4$

3. Add and subtract this value (4) inside the equation: $f(x) = (x^2 + 4x + 4) + 10 - 4$ $f(x) = (x + 2)^2 + 6$

So, the vertex form is: $f(x) = (x + 2)^2 + 6$

b) Is the graph concave up or concave down?

Since the coefficient of $x^2$ is positive (1), the parabola opens upwards. Therefore, the graph is concave up.

c) What is the y-intercept?

To find the y-intercept, we set $x = 0$ in the original function: $f(0) = 0^2 + 4(0) + 10 = 10$ Thus, the y-intercept is $(0, 10)$.

d) What is the vertex?

From the vertex form $f(x) = (x + 2)^2 + 6$, the vertex is $(-2, 6)$.

Would you like any more details on these steps or further clarification?

Here are 5 related questions for further exploration:

1. How do you determine the axis of symmetry from a quadratic function?
2. What is the significance of the vertex in a quadratic function's graph?
3. How does the value of $a$ in $ax^2 + bx + c$ affect the shape of the parabola?
4. Can you explain the process of completing the square in more detail?
5. How can you find the x-intercepts of a quadratic function?

Tip: The vertex form of a quadratic function easily reveals both the direction of the parabola and its maximum or minimum point.