Question

ticket 1
graph the equation.
$y = \frac{1}{2}x + 3$

ticket 2
graph the equation.
$y = -\frac{1}{2}x + 4$

ticket 3
graph the equation.
$y = 2x - 3$

ticket 4
graph the equation.
$x = 1$

Explanation:

Ticket 1: Graph \( y = \frac{1}{2}x + 3 \)

Step 1: Identify Slope and Intercept

The equation is in slope - intercept form \( y=mx + b \), where \( m=\frac{1}{2} \) (slope) and \( b = 3 \) (y - intercept).

Step 2: Plot the Y - Intercept

The y - intercept is \( 3 \), so plot the point \( (0,3) \) on the y - axis.

Step 3: Use the Slope to Find Another Point

The slope \( m=\frac{1}{2}=\frac{\text{rise}}{\text{run}} \). From the point \( (0,3) \), move up 1 unit (rise) and then 2 units to the right (run). This gives the point \( (2,4) \).

Step 4: Draw the Line

Draw a straight line through the points \( (0,3) \) and \( (2,4) \).

Ticket 2: Graph \( y=-\frac{1}{2}x + 4 \)

Step 1: Identify Slope and Intercept

In the slope - intercept form \( y = mx + b \), \( m=-\frac{1}{2} \) and \( b = 4 \).

Step 2: Plot the Y - Intercept

Plot the point \( (0,4) \) on the y - axis.

Step 3: Use the Slope to Find Another Point

The slope \( m =-\frac{1}{2}=\frac{\text{rise}}{\text{run}} \). Since the slope is negative, we can move down 1 unit (rise) and 2 units to the right (run) from \( (0,4) \), getting the point \( (2,3) \), or up 1 unit and 2 units to the left, getting \( (- 2,5) \).

Step 4: Draw the Line

Draw a straight line through the points (e.g., \( (0,4) \) and \( (2,3) \)).

Ticket 3: Graph \( y = 2x-3 \)

Step 1: Identify Slope and Intercept

For \( y=mx + b \), \( m = 2 \) and \( b=-3 \).

Step 2: Plot the Y - Intercept

Plot the point \( (0,-3) \) on the y - axis.

Step 3: Use the Slope to Find Another Point

The slope \( m = 2=\frac{2}{1}=\frac{\text{rise}}{\text{run}} \). From \( (0,-3) \), move up 2 units and 1 unit to the right, getting the point \( (1,-1) \).

Step 4: Draw the Line

Draw a straight line through \( (0,-3) \) and \( (1,-1) \).

Ticket 4: Graph \( x = 1 \)

Step 1: Analyze the Equation

The equation \( x = 1 \) is a vertical line. A vertical line has an undefined slope and passes through all points with an x - coordinate of 1.

Step 2: Plot Points and Draw the Line

Plot points such as \( (1,0) \), \( (1,1) \), \( (1,-1) \) etc. Then draw a vertical line through these points.

Answer:

The graphs are drawn as described above for each ticket. For Ticket 1: Line through (0,3) and (2,4); Ticket 2: Line through (0,4) and (2,3); Ticket 3: Line through (0, - 3) and (1,-1); Ticket 4: Vertical line through x = 1.