https://file.aiquickdraw.com/mathbot/file/17265856290753nzudct1.jpg

Let's break down the problem.

### Function:
You are given the quadratic function:

\[
f(x) = 2x^2 + 1
\]

### 1. **Graphing the Function**:
The function \( f(x) = 2x^2 + 1 \) is a parabola, and since the coefficient of \( x^2 \) is positive (2), the parabola opens upwards.

- The **vertex** of the parabola is at \( (0, 1) \), as the constant term 1 represents the y-coordinate of the vertex.
- The parabola is vertically stretched because of the factor of 2 in front of \( x^2 \), making it narrower than the standard parabola \( x^2 \).
- For points near the vertex:
    - At \( x = 0 \), \( f(0) = 2(0)^2 + 1 = 1 \).
    - At \( x = 1 \), \( f(1) = 2(1)^2 + 1 = 3 \).
    - At \( x = -1 \), \( f(-1) = 2(-1)^2 + 1 = 3 \).
    - At \( x = 2 \), \( f(2) = 2(2)^2 + 1 = 9 \).
    - At \( x = -2 \), \( f(-2) = 2(2)^2 + 1 = 9 \).

You can plot these points: \( (0, 1) \), \( (1, 3) \), \( (-1, 3) \), \( (2, 9) \), and \( (-2, 9) \) on the provided grid.

### 2. **Domain**:
The domain of the quadratic function \( f(x) = 2x^2 + 1 \) is all real numbers since the function is defined for all \( x \).

\[
\text{Domain}: (-\infty, \infty)
\]

### 3. **Range**:
The minimum value of the function occurs at the vertex \( (0, 1) \), and since the parabola opens upwards, the range is all values of \( y \geq 1 \).

\[
\text{Range}: [1, \infty)
\]

Would you like me to clarify any specific steps or plot the graph with further detail?

Here are 5 related questions:
1. How do you find the vertex of a quadratic function?
2. What impact does the coefficient in front of \( x^2 \) have on the shape of the parabola?
3. How would the graph change if the equation were \( f(x) = -2x^2 + 1 \)?
4. What is the axis of symmetry for the function \( f(x) = 2x^2 + 1 \)?
5. How would you determine the y-intercept and x-intercepts of this function?

**Tip**: For quadratic functions, the vertex form \( f(x) = a(x-h)^2 + k \) can help identify the vertex and the direction in which the parabola opens easily.

Graph the function f(x) = 2x^2 + 1 and determine the domain and range (answer in interval notation).

Learn how to graph the quadratic function f(x) = 2x^2 + 1, and explore its domain and range. This problem covers quadratic functions, graphing parabolas, and finding the domain and range in interval notation. Suitable for students in Grades 9-10.

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