https://file.aiquickdraw.com/mathbot/file/17258279037006q2zblwd.jpg

https://file.aiquickdraw.com/mathbot/file/172582790370179tfaxc3.jpg

Let's analyze the two images you've uploaded.

### First Image:
- The problem asks for different characteristics of the function displayed in the graph:
  - **(A) Intercepts**: Find where the graph intersects the x-axis and y-axis.
  - **(B) Vertex**: Identify the highest or lowest point of the parabola.
  - **(C) Maximum or Minimum**: Determine whether the vertex is a maximum (if the parabola opens downward) or minimum (if it opens upward).
  - **(D) Range**: Specify the set of y-values that the function takes.

The **x-intercepts** are the points where the graph crosses the x-axis. Looking at the graph:
- The graph crosses the x-axis at approximately \( x = -5 \) and \( x = 3 \) (rounded to the nearest integer).

The **y-intercept** is where the graph crosses the y-axis, which happens around \( y = -10 \).

The **vertex** is the highest point in this case because the parabola opens downward. The vertex appears to be located at \( (x, y) = (-1, 4) \).

Since the parabola opens downward, the vertex represents a **maximum** point.

Finally, the **range** of the function, considering the maximum point and the fact that the parabola extends downward, would be all values from \( -\infty \) to the y-coordinate of the vertex, \( 4 \). So, the range is \( (-\infty, 4] \).

### Second Image:
This problem asks you to match equations with their corresponding graph (denoted by f, g, m, and n).

For **y = -(x - 2)^2 + 3**, the vertex is at \( (2, 3) \), and since the coefficient of the squared term is negative, the parabola opens downward. Looking at the graphs, the one labeled **g** matches this description because its vertex is at \( (2, 3) \) and it opens downward.

Would you like to dive deeper into any part of these questions or explanations? 

Here are some related questions to expand understanding:
1. What are the steps to find the intercepts of a quadratic function?
2. How do you determine whether a parabola has a maximum or minimum point?
3. How does changing the coefficient in front of the squared term affect the graph's direction?
4. What is the relationship between the vertex form of a quadratic equation and the graph’s vertex?
5. How do you calculate the range of a quadratic function?

**Tip:** To find the x-intercepts of any quadratic function, set \( y = 0 \) and solve the equation for \( x \).

Analyze the graph and equations of quadratic functions to determine intercepts, vertices, maximum/minimum values, and match equations to their corresponding graphs.

Learn how to analyze quadratic function graphs to find intercepts, vertices, maximum/minimum points, and match equations with graphs. This detailed guide explores quadratic function graphing, including x-intercepts, y-intercepts, and identifying the vertex. Ideal for students in Grades 9-11.

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