We are asked to use the method of completing the square to find the standard form of the quadratic function, identify the axis of symmetry and the vertex, and sketch the graph.

The given quadratic function is:

$f(x) = x^2 + 4x + 1$

Step 1: Completing the Square

To convert $f(x) = x^2 + 4x + 1$ into its standard form, $f(x) = a(x-h)^2 + k$, we need to complete the square:

1. Start with the first two terms: $x^2 + 4x$.
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 4 within the equation:

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

$f(x) = (x + 2)^2 - 3$

So the standard form of the function is:

$y = (x + 2)^2 - 3$

Step 2: Vertex and Axis of Symmetry

The standard form is $y = (x-h)^2 + k$, where $(h, k)$ is the vertex and the axis of symmetry is $x = h$.

From the equation $y = (x + 2)^2 - 3$, we can see that:

• The vertex is $(-2, -3)$.
• The axis of symmetry is $x = -2$.

Summary:

• Standard form: $y = (x + 2)^2 - 3$
• Axis of symmetry: $x = -2$
• Vertex: $(-2, -3)$

Step 3: Sketching the Graph

• The graph is a parabola opening upwards because the coefficient of $(x + 2)^2$ is positive.
• The vertex $(-2, -3)$ is the lowest point on the graph.
• The axis of symmetry is the vertical line $x = -2$.

Would you like a detailed graph or further explanation on any of these steps?

Here are 5 questions to expand on the above:

1. How does completing the square help in finding the vertex of a quadratic function?
2. What is the significance of the axis of symmetry in a quadratic graph?
3. Why does the parabola open upwards in this function?
4. How can the vertex form be used to quickly sketch a quadratic graph?
5. How would you calculate the y-intercept of this function using the standard form?

Tip: When completing the square, always ensure you adjust the constant outside the parentheses to maintain equality in the equation.