Let's go through the questions from the image:

1. Graphing the parabola: The equation given is $y = -\frac{1}{4}(x - 3)^2 - 1$. This is in vertex form, where $(h, k)$ is the vertex. Here:

   • Vertex $(h, k) = (3, -1)$
   • The parabola opens downwards because of the negative sign in front of the quadratic term, and it's vertically compressed by a factor of $\frac{1}{4}$. You can plot the vertex at $(3, -1)$ and the points around it using symmetry.

2. Height function for jumping shoes: This question involves determining the maximum height for two given functions:

   • Max Jumps: $f(x) = -19.2(x - 2.539)^2 + 16$
   • Jumpsters: $g(x) = -19.2(x - 0.4457)^2 + 38$ Both functions are in vertex form, so the vertex gives us the maximum height for each.
   • Max Jumps: maximum height is 16 inches at $x = 2.539$ seconds.
   • Jumpsters: maximum height is 38 inches at $x = 0.4457$ seconds. Therefore, Beth would likely choose Jumpsters, as they allow a higher jump.

3. Finding additional points on a parabola: The parabola has vertex $(1, -2)$ and passes through $(0, -5)$. The equation of the parabola can be written in vertex form $y = a(x - h)^2 + k$, and we need to solve for $a$.

   • Vertex: $h = 1$, $k = -2$
   • Point $(0, -5)$: Plugging this into the equation to solve for $a$: $-5 = a(0 - 1)^2 - 2 \implies -5 = a(1) - 2 \implies a = -3$ Thus, the equation is $y = -3(x - 1)^2 - 2$. Using this equation, we can calculate the values for the given points and find which ones lie on the parabola.

4. Equation of parabola and converting to standard form: You are given the vertex $(4, 3)$ and another point $(3, -1)$. To find the equation in vertex form, use the format $y = a(x - h)^2 + k$.

   • Vertex: $(h, k) = (4, 3)$
   • Substituting $(3, -1)$ into the equation: $-1 = a(3 - 4)^2 + 3 \implies -1 = a(1)^2 + 3 \implies a = -4$ The vertex form of the equation is $y = -4(x - 4)^2 + 3$.

   To convert to standard form, expand the vertex form: $y = -4(x - 4)^2 + 3 \implies y = -4(x^2 - 8x + 16) + 3 \implies y = -4x^2 + 32x - 64 + 3 \implies y = -4x^2 + 32x - 61$

Would you like further details on any specific part or have any questions?

Here are 5 related questions:

1. How do you graph a parabola using the vertex and the value of $a$?
2. What does the value of $a$ indicate in the vertex form of a parabola?
3. How do you convert from vertex form to factored form for quadratic equations?
4. What is the importance of the vertex in determining the maximum or minimum value of a parabola?
5. How do changes in the equation's parameters affect the shape and position of the parabola?

Tip: When graphing parabolas, remember that the axis of symmetry passes through the vertex, making it easier to plot symmetric points.