Here is the solution for each question:

1. The graph of the parabola $f(x) = 3(x - 4)^2 + 5$ opens upward

• The parabola opens upward if the coefficient of $(x - h)^2$ is positive.
• Here, the coefficient is $3$, which is positive.
• Answer: True (A)

2. What is the axis of symmetry of $y = 3x^2 - 6x + 4$?

• The axis of symmetry for a parabola in the form $y = ax^2 + bx + c$ is given by: $x = -\frac{b}{2a}$
• Here, $a = 3$, $b = -6$, so: $x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1$
• Answer: $x = 1$ (B)

3. If $y = x^2 - 6x + 4$, find the vertex

• The x-coordinate of the vertex is the same as the axis of symmetry: $x = -\frac{b}{2a} = 1$
• Substitute $x = 1$ into $y = x^2 - 6x + 4$: $y = (1)^2 - 6(1) + 4 = 1 - 6 + 4 = -1$
• The vertex is $(1, -1)$.
• Answer: Not listed (vertex is $(1, -1)$).

4. Solve $x^2 - 4x < 5$

• Rearrange as $x^2 - 4x - 5 < 0$.
• Factorize: $(x - 5)(x + 1) < 0$.
• Solve the inequality: $x \in (-1, 5)$
• Answer: $(-1, 5)$ (B)

5. Find the equation of the parabola with vertex $(3, 2)$ and x-intercept $4$

• Vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
• Substitute $(h, k) = (3, 2)$: $y = a(x - 3)^2 + 2$
• Use the x-intercept $(4, 0)$: $0 = a(4 - 3)^2 + 2 \implies 0 = a(1)^2 + 2 \implies a = -2$
• Equation: $y = -2(x - 3)^2 + 2$.
• Expand to match options: $y = -2(x^2 - 6x + 9) + 2 = -2x^2 + 12x - 18 + 2 = -2x^2 + 12x - 16$
• Answer: $y = -2x^2 + 12x - 16$ (A)

Let me know if you need detailed steps for any solution! Here are related questions to deepen your understanding:

1. What is the general formula for determining the vertex of a quadratic function in standard form?
2. How do you determine whether a parabola opens upward or downward?
3. Can you explain how to factorize quadratic expressions in inequalities?
4. What is the significance of the axis of symmetry in the graph of a parabola?
5. How do you derive the vertex form of a parabola from its general equation?

Tip: Always double-check calculations for $a, b,$ and $c$ in quadratic functions to avoid simple algebraic errors.