The image you have uploaded contains twelve quadratic functions, where the task is to graph the functions and label the vertex and axis of symmetry.

Quadratic functions are generally of the form $f(x) = ax^2 + bx + c$. To find the vertex of a quadratic function in this form, we use the formula for the x-coordinate of the vertex:

$x = \frac{-b}{2a}$

The corresponding y-coordinate is found by substituting this value back into the function. The axis of symmetry is the vertical line $x = \frac{-b}{2a}$.

Let's analyze the first three functions as an example:

1. $f(x) = -3x^2 - 2x - 4$

• a = -3, b = -2, c = -4.
• The x-coordinate of the vertex: $x = \frac{-(-2)}{2(-3)} = \frac{2}{-6} = -\frac{1}{3}$
• Substituting $x = -\frac{1}{3}$ into the equation to find the y-coordinate of the vertex: $f\left( -\frac{1}{3} \right) = -3\left( -\frac{1}{3} \right)^2 - 2\left( -\frac{1}{3} \right) - 4$ Simplifying: $f\left( -\frac{1}{3} \right) = -3\left( \frac{1}{9} \right) + \frac{2}{3} - 4 = -\frac{1}{3} + \frac{2}{3} - 4 = -\frac{1}{3} + \frac{2}{3} - \frac{12}{3} = -\frac{11}{3}$
• So the vertex is $\left( -\frac{1}{3}, -\frac{11}{3} \right)$, and the axis of symmetry is $x = -\frac{1}{3}$.

2. $f(x) = 3(x - 1)^2 + 5$

This is already in vertex form, $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• The vertex is $(1, 5)$.
• The axis of symmetry is $x = 1$.

3. $g(x) = -\frac{1}{2}(x + 3)^2 + 2$

This is also in vertex form.

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

Would you like me to proceed with the rest of the functions or focus on one particular function? Let me know if you want detailed steps on how to graph these as well.

Here are 5 questions related to this information:

1. How does the sign of $a$ affect the direction of the parabola's opening?
2. What is the difference between the standard form and vertex form of a quadratic function?
3. How can the axis of symmetry help in graphing quadratic functions?
4. Can a quadratic function have a vertex that lies on the x-axis? If so, what does it indicate?
5. How do transformations of the parent function $f(x) = x^2$ affect the graph of a quadratic function?

Tip: In vertex form, $a$ controls the width and direction of the parabola, while $(h, k)$ directly gives you the vertex without needing further calculations.