https://file.aiquickdraw.com/mathbot/file/1729549246488zqw06m3b.jpg

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.

The user uploaded an image with questions involving quadratic functions in vertex form, solving for the equation of a parabola, and analyzing maximum height functions.

Practice solving quadratic functions in vertex form, graphing parabolas, and determining the maximum height of jumping functions. Ideal for high school students in Grades 9-12.

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