Question

question 4 (3 points)
select the correct graph for the function \\(f(x) = 3x + 4\\).

a)
b)
c)
d)

Explanation:

Identify the key features of the linear function

The given function is:

$$ f(x) = 3x + 4 $$

This is in slope-intercept form \(y = mx + b\), where:

• The slope \(m = 3\)
• The \(y\)-intercept \(b = 4\), corresponding to the point \((0, 4)\)

Determine key points on the graph

Using the function \(f(x) = 3x + 4\):

• When \(x = 0\), \(y = 4\). The graph must cross the \(y\)-axis at \((0, 4)\).
• When \(x = -1\), \(y = 3(-1) + 4 = 1\). The point \((-1, 1)\) is on the line.
• When \(x = -2\), \(y = 3(-2) + 4 = -2\). The point \((-2, -2)\) is on the line.
• When \(x = -3\), \(y = 3(-3) + 4 = -5\). The point \((-3, -5)\) is on the line.
• When \(x = -4\), \(y = 3(-4) + 4 = -8\). The point \((-4, -8)\) is on the line.

Match with the given options

• Option A: The line passes through \((0, 4)\), \((-1, 1)\), \((-2, -2)\), \((-3, -5)\), and \((-4, -8)\). This matches our calculated points perfectly.
• Option B: The line has a \(y\)-intercept at \((0, 9)\), which is incorrect.
• Option C: The line is extremely steep and has a different intercept.
• Option D: The line has a positive \(x\)-intercept and a negative \(y\)-intercept, which is incorrect.

Answer:

• (A) A line passing through the y-intercept (0, 4) with a slope of 3 (Correct answer)
• (B) A line passing through the y-intercept (0, 9)
• (C) A nearly vertical line with a different intercept
• (D) A line with a negative y-intercept