Determine if the relation y=3x+1 is a function by first graphing the relation, then applying the vertical line test. Is this equation a function, yes, or no?

Answer 1

To determine if the relation \( y = 3x + 1 \) is a function, we can analyze it in two ways: graphing the equation and applying the vertical line test.

1. Graphing the equation: The equation \( y = 3x + 1 \) is a linear equation in slope-intercept form, where the slope (m) is 3 and the y-intercept (b) is 1. This means that the graph is a straight line that crosses the y-axis at the point (0, 1) and has a steep incline.

   • If you plot points using different x-values:
     • For \( x = 0 \), \( y = 3(0) + 1 = 1 \) → point (0, 1)
     • For \( x = 1 \), \( y = 3(1) + 1 = 4 \) → point (1, 4)
     • For \( x = -1 \), \( y = 3(-1) + 1 = -2 \) → point (-1, -2)

   The resulting graph will display a straight line passing through these points.

2. Applying the Vertical Line Test: The vertical line test states that a relation is a function if every vertical line intersects the graph at most once.

   Since the graph of \( y = 3x + 1 \) is a straight line, any vertical line will only intersect the graph at one point. Therefore, the vertical line test confirms that it is indeed a function.

In conclusion, since both the graphing and the vertical line test verify that the relation \( y = 3x + 1 \) passes the test of being a function, we can confidently say: Yes, this equation is a function.