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The equation given is:

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

This is a quadratic function, so let's break it down step by step.

a) Graph the function $f(x) = 2(x + 3)^2$

This equation is in vertex form:

$f(x) = a(x-h)^2 + k$

where $(h, k)$ is the vertex of the parabola, and $a$ determines the direction and the width of the parabola. In this case:

• $a = 2$, so the parabola opens upwards and is vertically stretched by a factor of 2.
• $h = -3$ and $k = 0$, so the vertex is at $(-3, 0)$.

The general shape of this graph is a parabola that opens upwards, with its vertex shifted to the left by 3 units. The scaling factor of 2 makes it narrower than a standard parabola.

b) Identify the vertex

From the equation, we can directly identify the vertex:

$\text{Vertex} = (-3, 0)$

c) Identify the axis of symmetry

The axis of symmetry for any parabola is the vertical line that passes through the vertex. So for this function:

$x = -3$

This is the equation for the axis of symmetry.

Would you like me to create a plot for the graph? Also, let me know if you have any questions on these parts! Here are 5 related questions:

1. What is the impact of the coefficient $a$ in a parabola's equation?
2. How would the graph change if the equation were $f(x) = -2(x + 3)^2$?
3. How do you find the x-intercepts for a quadratic equation?
4. What is the general form of a quadratic equation, and how is it different from the vertex form?
5. Can you explain how to calculate the y-intercept for the function $f(x) = 2(x + 3)^2$?

Tip: The vertex form of a quadratic equation is extremely useful for quickly identifying the vertex and the direction of the parabola.