https://file.aiquickdraw.com/mathbot/file/1727458265214w8uury5a.jpg

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 \( (-2, 6) \).

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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.

Analyze the function f(x) = x^2 + 4x + 10. 
a) Rewrite f(x) in vertex form. 
b) Is the graph of the function concave up or concave down? 
c) What is the y-intercept? 
d) What is the vertex?

Learn how to rewrite the quadratic function f(x) = x^2 + 4x + 10 in vertex form, determine its concavity, find the y-intercept, and identify the vertex. This problem introduces key concepts in quadratic functions and graphing, suitable for Grades 9-10.

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