Math Problem Statement

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=​?

...

Question content area top right

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.

Question content area bottom

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

Solution

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)(1, -4), the lowest y-value is −4-4 and it extends infinitely upwards.

Range: [−4,∞)[-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)(-1, 0) and (3,0)(3, 0).

x-intercepts: −1,3-1, 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)(0, -3).

y-intercept: −3-3

e. Missing function values

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

  • f(−2)=5f(-2) = 5
  • f(2)=−3f(2) = -3

So, the required values are:

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

Summary

a. Domain: (−∞,∞)(-\infty, \infty)

b. Range: [−4,∞)[-4, \infty)

c. x-intercepts: −1,3-1, 3

d. y-intercept: −3-3

e. f(−2)=5f(-2) = 5 and f(2)=−3f(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.

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Math Problem Analysis

Mathematical Concepts

Functions
Graphing
Quadratic Equations

Formulas

-

Theorems

-

Suitable Grade Level

Grades 9-12

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