Let's go through each quadratic function, finding the axis of symmetry, vertex, y-intercept, maximum or minimum point, and the domain and range.

Function 1: $f(x) = 3x^2 - 6x + 1$

1. Axis of Symmetry: The axis of symmetry for a quadratic function $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$.

   • Here, $a = 3$ and $b = -6$.
   • So, $x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1$.

2. Vertex: Substitute $x = 1$ into the function to find the y-coordinate of the vertex.

   • $f(1) = 3(1)^2 - 6(1) + 1 = 3 - 6 + 1 = -2$.
   • Vertex: $(1, -2)$.

3. Y-intercept: The y-intercept is the value of $f(x)$ when $x = 0$.

   • $f(0) = 3(0)^2 - 6(0) + 1 = 1$.
   • Y-intercept: $(0, 1)$.

4. Maximum or Minimum: Since $a = 3 > 0$, this parabola opens upwards, so it has a minimum at the vertex.

   • Minimum point: $(1, -2)$.

5. Domain and Range:

   • Domain: All real numbers, $(-\infty, \infty)$.
   • Range: $[-2, \infty)$, since the minimum value is -2.

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

1. Axis of Symmetry:

   • $a = -1$ and $b = -2$.
   • $x = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1$.

2. Vertex:

   • Substitute $x = -1$ into the function.
   • $f(-1) = -(-1)^2 - 2(-1) + 3 = -1 + 2 + 3 = 4$.
   • Vertex: $(-1, 4)$.

3. Y-intercept:

   • $f(0) = -0^2 - 2(0) + 3 = 3$.
   • Y-intercept: $(0, 3)$.

4. Maximum or Minimum: Since $a = -1 < 0$, this parabola opens downwards, so it has a maximum at the vertex.

   • Maximum point: $(-1, 4)$.

5. Domain and Range:

   • Domain: All real numbers, $(-\infty, \infty)$.
   • Range: $(-\infty, 4]$, since the maximum value is 4.

Function 3: $f(x) = x^2 - 4$

1. Axis of Symmetry:

   • $a = 1$ and $b = 0$.
   • $x = -\frac{0}{2 \cdot 1} = 0$.

2. Vertex:

   • Substitute $x = 0$ into the function.
   • $f(0) = (0)^2 - 4 = -4$.
   • Vertex: $(0, -4)$.

3. Y-intercept:

   • $f(0) = -4$.
   • Y-intercept: $(0, -4)$.

4. Maximum or Minimum: Since $a = 1 > 0$, this parabola opens upwards, so it has a minimum at the vertex.

   • Minimum point: $(0, -4)$.

5. Domain and Range:

   • Domain: All real numbers, $(-\infty, \infty)$.
   • Range: $[-4, \infty)$, since the minimum value is -4.

Would you like more detailed explanations for any of these steps, or do you have further questions?

Additional Related Questions

1. How do you determine if a quadratic function has a maximum or minimum point?
2. How can you find the x-intercepts of a quadratic function?
3. What is the significance of the axis of symmetry in a quadratic function?
4. How can transformations affect the graph of a quadratic function?
5. How do the values of $a$, $b$, and $c$ in $ax^2 + bx + c$ affect the shape and position of the graph?

Tip

Remember that the vertex of a quadratic function $ax^2 + bx + c$ provides the point where the function reaches its maximum or minimum, depending on whether the parabola opens up or down.