Question

the equation y = 3x + 12 represents the total cost, y, of x hot dogs and a souvenir hat from the concession stand at a football game. graph this equation. (use desmos)

Explanation:

1. Slope - Intercept Form: The equation of a line in slope - intercept form is \( y=mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept (the value of \( y \) when \( x = 0\)). In the context of the problem, \( y \) is the total cost, \( x \) is the number of hot dogs, the slope represents the cost per hot dog, and the y - intercept represents the cost of the souvenir hat (a fixed cost).
2. Plotting Points: We first plot the y - intercept because it gives us a starting point on the y - axis. Then, using the slope (which tells us how much \( y \) changes for a unit change in \( x \)), we can find another point on the line.
3. Drawing the Line: A straight line is drawn through the plotted points because the equation is a linear equation, and the graph of a linear equation is a straight line.

Answer:

To graph the equation \( y = 3x + 12 \) (assuming the original equation has a typo and should be \( y = 3x + 12 \) instead of \( y = 1x + 12 \) for a more meaningful cost context, or if it's \( y = x + 12 \), the process is similar):

Step 1: Identify the slope and y - intercept

The equation is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

• If the equation is \( y = 3x+12 \), then the slope \( m = 3\) and the y - intercept \( b = 12\).
• If the equation is \( y=x + 12\), then the slope \( m = 1\) and the y - intercept \( b = 12\).

Step 2: Plot the y - intercept

The y - intercept is the point where \( x = 0\). Substitute \( x = 0\) into the equation:

• For \( y=3x + 12\), when \( x = 0\), \( y=3(0)+12=12\). So we plot the point \( (0,12) \).
• For \( y=x + 12\), when \( x = 0\), \( y=0 + 12=12\). So we plot the point \( (0,12) \).

Step 3: Use the slope to find another point

The slope \( m=\frac{\text{rise}}{\text{run}} \).

• If \( m = 3=\frac{3}{1}\), from the point \( (0,12) \), we rise 3 units (move up 3) and run 1 unit (move right 1). So we get the point \( (0 + 1,12+3)=(1,15) \).
• If \( m = 1=\frac{1}{1}\), from the point \( (0,12) \), we rise 1 unit (move up 1) and run 1 unit (move right 1). So we get the point \( (0 + 1,12 + 1)=(1,13) \).

Step 4: Draw the line

Draw a straight line through the two (or more) points we have plotted. This line represents the equation \( y = 3x+12 \) (or \( y=x + 12\)) on the coordinate plane.

If we use Desmos (as suggested in the problem), we can simply type the equation \( y = 3x + 12 \) (or the correct equation) into the Desmos graphing calculator, and it will plot the line for us. The line will have a positive slope (since the coefficient of \( x \) is positive) and will cross the y - axis at \( y = 12\).

(Note: If the original equation is indeed \( y=x + 12\), the steps are analogous, with the slope being 1 instead of 3. The key is to recognize the slope - intercept form and use it to graph the line.)