Question

which equation represents the line shown in the graph?
a. $y = 4x - 3$
b. $y = 3x + 4$
c. $y = 4x + 3$
d. $y = 3x - 4$

Explanation:

Step1: Identify the y - intercept

The y - intercept (\(b\)) is the point where the line crosses the y - axis. From the graph, the line crosses the y - axis at \(y = 4\), so \(b=4\). The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. So the equation of the line should have \(+ 4\) as the constant term. Let's check the options:

• Option A: \(y = 4x-3\), \(b=-3\) (incorrect)
• Option B: \(y = 3x + 4\), \(b = 4\) (possible)
• Option C: \(y=4x + 3\), \(b = 3\) (incorrect)
• Option D: \(y=3x-4\), \(b=-4\) (incorrect)

Step2: Calculate the slope

We can calculate the slope (\(m\)) using two points on the line. We know the y - intercept is \((0,4)\). Let's take another point. From the graph, when \(x = 1\), \(y=3\times1 + 4=7\)? Wait, no, let's use the two - point formula. Let's take the points \((0,4)\) and \((1,7)\)? Wait, no, let's use the points \((0,4)\) and \((-1,1)\)? Wait, no, let's calculate the slope between \((0,4)\) and \((1,7)\)? Wait, no, let's use the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: when \(x = 0\), \(y = 4\) (point \((0,4)\)) and when \(x = 1\), let's see the graph. The line goes up 3 units for every 1 unit to the right. So the slope \(m = 3\). So the equation of the line is \(y=3x + 4\), which is option B.

Answer:

B. \(y = 3x + 4\)