Question

1. $y = 3x + 1$ (and a coordinate grid chart)

Explanation:

To graph \( y = 3x + 1 \), we can use the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step 1: Identify the y - intercept

For the equation \( y = 3x+1 \), when \( x = 0 \), we substitute \( x = 0 \) into the equation:
\( y=3(0)+1=1 \). So the y - intercept is \( (0,1) \). We can plot this point on the graph (where the x - coordinate is 0 and the y - coordinate is 1).

Step 2: Use the slope to find another point

The slope \( m = 3=\frac{3}{1} \), which means for every 1 unit we move to the right (increase in \( x \) by 1), we move up 3 units (increase in \( y \) by 3).
Starting from the y - intercept \( (0,1) \), if we move \( x = 1 \) (to the right 1 unit), then \( y=3(1)+1 = 4 \). So the point \( (1,4) \) is also on the line.
We can also find a point by moving left. If \( x=- 1 \), then \( y=3(-1)+1=-3 + 1=-2 \), so the point \( (-1,-2) \) is on the line.

Step 3: Draw the line

Once we have two or more points (e.g., \( (0,1) \), \( (1,4) \), \( (-1,-2) \)) plotted on the coordinate plane, we can draw a straight line passing through these points.

If we were to find the value of \( y \) for a given \( x \), say \( x = 2 \):

Step1: Substitute \( x = 2 \) into the equation

\( y=3(2)+1 \)

Step2: Simplify the expression

\( y = 6 + 1=7 \)

If the question was to graph the line \( y = 3x+1 \), the key points are \( (0,1) \), \( (1,4) \), \( (-1,-2) \) and we draw a straight line through them. If it was to find \( y \) for a particular \( x \), we use substitution as shown above.

If we assume the question is to graph the line, the process is as described. If we take a sample calculation, for \( x = 0 \), \( y = 1 \); for \( x = 1 \), \( y = 4 \); for \( x=-1 \), \( y=-2 \) and then draw the line.

If the problem was to find the slope and y - intercept:
The equation is in slope - intercept form \( y=mx + b \), where \( m = 3 \) (slope) and \( b = 1 \) (y - intercept).

Answer:

To graph \( y = 3x + 1 \), plot points like \((0,1)\), \((1,4)\), \((-1,-2)\) and draw a straight line. (If calculating \( y \) for a specific \( x \), follow the substitution steps above. For slope - intercept form analysis, slope \( m = 3 \), y - intercept \( b = 1 \))