https://file.aiquickdraw.com/mathbot/file/1726358020465ksue2m7e.jpg

Let's analyze the linear function \( f(x) = \frac{2}{3}x + 4 \) and answer the questions from the image.

### a) Integer coordinates of the y-intercept
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:

\[
f(0) = \frac{2}{3}(0) + 4 = 4
\]

So, the y-intercept is \( (0, 4) \).

### b) Integer coordinates of the x-intercept
The x-intercept occurs when \( f(x) = 0 \). Set the function equal to 0 and solve for \( x \):

\[
0 = \frac{2}{3}x + 4
\]
\[
\frac{2}{3}x = -4
\]
\[
x = -4 \times \frac{3}{2} = -6
\]

So, the x-intercept is \( (-6, 0) \).

### c) Domain of the function
The function is a linear function, which means it is defined for all real values of \( x \). Therefore, the domain is:

\[
(-\infty, \infty)
\]

### d) Range of the function
Since the function is linear and its graph is a straight line that extends infinitely in both directions, the range is also all real values:

\[
(-\infty, \infty)
\]

### e) Is the function increasing or decreasing?
The slope of the function is \( \frac{2}{3} \), which is positive. This means the function is increasing on its domain.

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Would you like any further details or explanations? Here are five related questions you might want to explore:

1. How do you graph linear functions on DESMOS?
2. What is the effect of the slope on the steepness of a linear graph?
3. How do you find the slope of a line given two points?
4. How do domain and range differ for different types of functions?
5. How does the behavior of a function change if the slope is negative?

**Tip**: Always check the sign of the slope to determine if a function is increasing or decreasing. A positive slope means the function increases, while a negative slope means it decreases.

Sketch the following linear function using DESMOS: f(x) = (2/3)x + 4. Analyze the graph by answering the following questions: a) State the integer coordinates of the y-intercept. b) State the integer coordinates of the x-intercept. c) State the interval for x representing the mathematical domain. d) State the interval for y representing the mathematical range. e) Is the function increasing or decreasing on its domain?

Learn how to analyze the linear function f(x) = (2/3)x + 4 by finding its x- and y-intercepts, determining its domain and range, and understanding its behavior. This exercise covers graphing and intercepts, suitable for students in Grades 9-11.

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