Math Problem Statement

Determine the domain, range, x-intercept, y-intercept, and equation of the line in the graph provided.

Solution

From the image provided, let's analyze the graph of the function and answer the questions.

Observations:

  • The graph is a straight line, indicating a linear function.
  • The line crosses the xx-axis (x-intercept) and the yy-axis (y-intercept).

Answers:

  1. Domain: For linear functions, the domain is all real numbers. Domain: (,)\text{Domain: } (-\infty, \infty)

  2. Range: Similarly, for linear functions, the range is also all real numbers. Range: (,)\text{Range: } (-\infty, \infty)

  3. Coordinates of the x-intercept: The x-intercept is where the line crosses the x-axis (y=0y=0). From the graph, the x-intercept is approximately at x=4x=4. Hence: Coordinates of the x-intercept: (4,0)\text{Coordinates of the x-intercept: } (4, 0)

  4. Coordinates of the y-intercept: The y-intercept is where the line crosses the y-axis (x=0x=0). From the graph, the y-intercept is approximately at y=2y=2. Hence: Coordinates of the y-intercept: (0,2)\text{Coordinates of the y-intercept: } (0, 2)

  5. The function:

    • From the graph, the slope (mm) of the line can be calculated. Between x=0x=0 and x=4x=4, the line decreases by 2 units vertically and increases by 4 units horizontally: m=ΔyΔx=24=12m = \frac{\Delta y}{\Delta x} = \frac{-2}{4} = -\frac{1}{2}
    • The y-intercept (bb) is 2. Using the slope-intercept form y=mx+by = mx + b: The function is: y=12x+2\text{The function is: } y = -\frac{1}{2}x + 2

Summary of Answers:

  1. Domain: (,)(-∞, ∞)
  2. Range: (,)(-∞, ∞)
  3. Coordinates of the x-intercept: (4,0)(4, 0)
  4. Coordinates of the y-intercept: (0,2)(0, 2)
  5. The function is: y=12x+2y = -\frac{1}{2}x + 2

Would you like a more detailed explanation on any of these points? Here are 5 additional questions for further exploration:

  1. How can you verify the slope using two points on the line?
  2. Why is the domain of a linear function always all real numbers?
  3. How would the function change if the slope were positive?
  4. Can we determine other points on the line using the equation?
  5. What does the y-intercept represent in real-world applications?

Tip: When working with a graph, always identify key points like intercepts and slopes to easily deduce the function's equation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Functions
Graph Interpretation

Formulas

Slope-Intercept Form: y = mx + b

Theorems

Properties of Linear Functions

Suitable Grade Level

Grades 8-10