Question

1. graph y = 3x + 1

Explanation:

Step1: Identify the equation type

The equation \( y = 3x + 1 \) is a linear equation in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept). Here, \( m = 3 \) and \( b=1 \).

Step2: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation \( y=3x + 1 \). We get \( y=3(0)+1=1 \). So, the point \( (0,1) \) is on the line.

Step3: Use the slope to find another point

The slope \( m = 3=\frac{3}{1} \), which means for a change of \( \Delta x = 1 \) (moving 1 unit to the right along the x - axis), \( \Delta y=3 \) (moving 3 units up along the y - axis). Starting from the y - intercept \( (0,1) \), if we move 1 unit to the right (\( x = 0 + 1=1 \)) and 3 units up (\( y=1 + 3 = 4 \)), we get the point \( (1,4) \). We can also find a point to the left of the y - intercept. For \( \Delta x=- 1 \) (moving 1 unit to the left), \( \Delta y=-3 \) (moving 3 units down). So, when \( x = 0-1=-1 \), \( y=1-3=-2 \), giving the point \( (-1,-2) \).

Step4: Plot the points and draw the line

Plot the points \( (0,1) \), \( (1,4) \), and \( (-1,-2) \) on the coordinate plane. Then, draw a straight line passing through these points.

(Note: If we consider the grid provided, we can use the two - point method or the slope - intercept method to sketch the line. The line should have a steep positive slope (since the slope is 3) and cross the y - axis at \( (0,1) \).)

Answer:

To graph \( y = 3x+1 \):

1. Plot the y - intercept at \( (0,1) \).
2. Use the slope \( m = 3 \) to find another point (e.g., from \( (0,1) \), move 1 unit right and 3 units up to get \( (1,4) \), or 1 unit left and 3 units down to get \( (-1,-2) \)).
3. Draw a straight line through the plotted points.