https://file.aiquickdraw.com/mathbot/file/1726703719050ex7wmy8z.jpg

Let's address the questions from the image. The task requires the following information for each quadratic function:

1. Opening of the graph (up or down).
2. Vertex of the parabola.
3. Domain.
4. Range.
5. Equation of the axis of symmetry.

We'll analyze the three quadratic functions one by one.

### 1. \( y = (x - 2)^2 + 3 \)
- **(a) Opening of the graph**: The coefficient of \( (x - 2)^2 \) is positive (1), so the parabola opens **upwards**.
- **(b) Vertex**: This is in vertex form \( y = (x - h)^2 + k \), where \( h = 2 \) and \( k = 3 \). Thus, the vertex is **(2, 3)**.
- **(c) Domain**: The domain of a quadratic function is always **all real numbers**: \( (-\infty, \infty) \).
- **(d) Range**: Since the parabola opens upwards and the minimum value is \( y = 3 \) (at the vertex), the range is \( [3, \infty) \).
- **(e) Axis of symmetry**: The axis of symmetry is the vertical line through the vertex, so \( x = 2 \).

### 2. \( y = -5(x - 2)^2 - 3 \)
- **(a) Opening of the graph**: The coefficient of \( (x - 2)^2 \) is negative (-5), so the parabola opens **downwards**.
- **(b) Vertex**: The vertex form is \( y = a(x - h)^2 + k \), where \( h = 2 \) and \( k = -3 \). The vertex is **(2, -3)**.
- **(c) Domain**: Again, the domain is **all real numbers**: \( (-\infty, \infty) \).
- **(d) Range**: Since the parabola opens downwards, the maximum value is \( y = -3 \) (at the vertex), so the range is \( (-\infty, -3] \).
- **(e) Axis of symmetry**: The axis of symmetry is \( x = 2 \).

### 3. \( y = 4(x + 3)^2 - 20 \)
- **(a) Opening of the graph**: The coefficient of \( (x + 3)^2 \) is positive (4), so the parabola opens **upwards**.
- **(b) Vertex**: The vertex form is \( y = a(x - h)^2 + k \), where \( h = -3 \) and \( k = -20 \). The vertex is **(-3, -20)**.
- **(c) Domain**: The domain is **all real numbers**: \( (-\infty, \infty) \).
- **(d) Range**: Since the parabola opens upwards and the minimum value is \( y = -20 \), the range is \( [-20, \infty) \).
- **(e) Axis of symmetry**: The axis of symmetry is \( x = -3 \).

Would you like more details on any of these? 

Here are five related questions for further understanding:
1. How do you convert a quadratic function into vertex form if it's not already?
2. What is the significance of the axis of symmetry in quadratic functions?
3. How does the coefficient \( a \) affect the shape of the parabola in terms of width?
4. What happens to the graph if the value of \( k \) in \( y = a(x - h)^2 + k \) changes?
5. How do you determine the range if you only have the vertex form?

**Tip**: In the quadratic function \( y = a(x - h)^2 + k \), the value of \( a \) controls the direction of the parabola's opening and its steepness. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

For each of the following quadratic functions, give the (a) opening of the graph, (b) vertex, (c) domain, (d) range, and (e) equation of the axis of symmetry.

Analyze quadratic functions to determine the graph's opening, vertex, domain, range, and the axis of symmetry. Explore key concepts in quadratic functions and their real-world applications, suitable for students in Grades 9-10.

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