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Use the graph to determine **a.** the​ function's domain; **b.** the​ function's range; **c.** the​ x-intercepts, if​ any; **d.** the​ y-intercept, if​ any; and **e.** the missing function​ values, indicated by question​ marks, below.

f left parenthesis negative 2 right parenthesisf(−2)equals=​? 
  
f left parenthesis 2 right parenthesisf(2)equals=​?

...

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Part 1

-55-55xy

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. A parabola that opens upward has vertex (1, negative 4) and passes through the points (negative 1, 0), (3, 0), left parenthesis 0 comma negative 3 right parenthesis comma left parenthesis negative 2 comma 5 right parenthesis and left parenthesis 2 comma negative 3 right parenthesis.

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Part 1

**a.** The domain is 

enter your response here.

 ​(Use interval​ notation.)

Part 2

**b.** The range is 

enter your response here.

​ (Use interval​ notation.)

Part 3

**c.** Select the correct choice below and fill in any answer boxes within your choice.

A.

The​ x-intercept(s) is​ (are) 

enter your response here.

​(Type an integer. Use a comma to separate answers as​ needed.)

B.

There is no​ x-intercept.

Part 4

**d.** Select the correct choice below and fill in any answer boxes within your choice.

A.

The​ y-intercept is 

enter your response here. 

​(Type an​ integer.)

B.

There is no​ y-intercept.

Part 5

**e.** 

f left parenthesis negative 2 right parenthesisf(−2)equals=enter your response here

Part 6

f left parenthesis 2 right parenthesisf(2)equals=enter your response here

Let's analyze the given graph of the parabola to determine the required values:

### a. Domain of the function

The domain of a function is the set of all possible x-values that the function can take. For this parabola, which extends infinitely in both the positive and negative directions along the x-axis:

**Domain**: \( (-\infty, \infty) \)

### b. Range of the function

The range of a function is the set of all possible y-values that the function can take. Since this is a parabola that opens upwards with the vertex at \((1, -4)\), the lowest y-value is \(-4\) and it extends infinitely upwards.

**Range**: \( [-4, \infty) \)

### c. x-intercepts

The x-intercepts are the points where the function intersects the x-axis. From the graph, we can see that the parabola intersects the x-axis at \((-1, 0)\) and \((3, 0)\).

**x-intercepts**: \(-1, 3\)

### d. y-intercept

The y-intercept is the point where the function intersects the y-axis. From the graph, we can see that the parabola intersects the y-axis at \((0, -3)\).

**y-intercept**: \(-3\)

### e. Missing function values

To find \( f(-2) \) and \( f(2) \), we can read the corresponding y-values from the graph for these x-values.

- \( f(-2) = 5 \)
- \( f(2) = -3 \)

So, the required values are:

**f(-2) = 5**  
**f(2) = -3**

### Summary
a. Domain: \( (-\infty, \infty) \)

b. Range: \( [-4, \infty) \)

c. x-intercepts: \(-1, 3\)

d. y-intercept: \(-3\)

e. \( f(-2) = 5 \) and \( f(2) = -3 \)

Would you like any further details or have any questions?

Here are some related questions you might find interesting:

1. What is the vertex form of the equation of the parabola?
2. How do you find the vertex of a parabola algebraically?
3. What are the axis of symmetry for the given parabola?
4. How do you convert a quadratic function from standard form to vertex form?
5. What is the significance of the y-intercept in a quadratic function?
6. How do you determine the direction in which a parabola opens?
7. How do you find the minimum or maximum value of a quadratic function?
8. What is the standard form equation of the given parabola?

**Tip:** When analyzing parabolas, always check the vertex and the direction in which the parabola opens to determine the range of the function accurately.

Learn how to analyze a parabola to determine its domain, range, x-intercepts, y-intercept, and function values at specific points. This problem involves graphing a parabola with a given vertex and points, covering key concepts in functions and quadratic equations. Suitable for students in Grades 9-12.

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