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Part 1
Graph the given​ functions, f and​ g, in the same rectangular coordinate system. Select integers for​ x, starting with​ -2 and ending with​ 2, and describe how the graph of g is related to the graph of f.
f left parenthesis x right parenthesis equals x squared​, g left parenthesis x right parenthesis equals x squared plus 1
Question content area bottom left
Part 1
Complete the ordered pairs in the table below for f left parenthesis x right parenthesis equals x squared.
x
​(x, f(x))
minus2
  
enter your response here
minus1
  
enter your response here
0
  
enter your response here
1
  
enter your response here
2
  
enter your response here
​(Simplify your answers. Type ordered​ pairs.

### Part 1: Graphing the functions \( f(x) = x^2 \) and \( g(x) = x^2 + 1 \)

1. **Function \( f(x) = x^2 \)**: This is a standard parabola that opens upwards, with its vertex at the origin \((0, 0)\).

2. **Function \( g(x) = x^2 + 1 \)**: This function is the graph of \( f(x) = x^2 \) shifted **upward by 1 unit**. The vertex of the parabola is at \((0, 1)\), but the shape of the graph remains identical to \( f(x) \).

### Part 2: Completing the table for \( f(x) = x^2 \)

Let's compute the values for \( f(x) = x^2 \) at the given points \( x = -2, -1, 0, 1, 2 \).

\[
f(x) = x^2
\]
- \( f(-2) = (-2)^2 = 4 \)
- \( f(-1) = (-1)^2 = 1 \)
- \( f(0) = 0^2 = 0 \)
- \( f(1) = 1^2 = 1 \)
- \( f(2) = 2^2 = 4 \)

### Completed Table:

| \( x \)  | \( (x, f(x)) \) |
|----------|-----------------|
| \( -2 \) | \( (-2, 4) \)    |
| \( -1 \) | \( (-1, 1) \)    |
| \( 0 \)  | \( (0, 0) \)     |
| \( 1 \)  | \( (1, 1) \)     |
| \( 2 \)  | \( (2, 4) \)     |

### Description of the relationship between \( f(x) \) and \( g(x) \):
The graph of \( g(x) = x^2 + 1 \) is the graph of \( f(x) = x^2 \), shifted **upward by 1 unit**. Both graphs are parabolas with the same shape, but \( g(x) \) has a vertex at \( (0, 1) \), while \( f(x) \) has a vertex at \( (0, 0) \).

---

Would you like me to go into more detail or answer any further questions?

Here are five additional questions to consider:

1. How does shifting a graph affect its equation?
2. What other transformations could be applied to \( f(x) = x^2 \)?
3. How does the vertex of a parabola change with vertical shifts?
4. What happens to the graph of \( f(x) \) if you subtract 1 instead of adding it?
5. How does reflecting a graph over the x-axis affect its equation?

**Tip:** To graph transformations easily, always focus on how shifts affect the vertex of the function.

Graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2, and describe how the graph of g is related to the graph of f. f(x) = x^2, g(x) = x^2 + 1. Complete the ordered pairs in the table below for f(x) = x^2. (Simplify your answers and type ordered pairs.)

Learn how to graph the quadratic functions f(x) = x^2 and g(x) = x^2 + 1 on the same coordinate plane. Understand how g(x) is a vertical shift of f(x) and complete a table of values for f(x). Suitable for students in Grades 8-10.

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