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To determine which graph represents the equation [tex]\( y = 12 + 2x \)[/tex] for a medium pizza with toppings, let's break down the components of the equation and what they mean in the context of the graph.
### Step-by-Step Solution:
1. Identify the Equation Format:
The equation [tex]\( y = 12 + 2x \)[/tex] is in the slope-intercept form of a linear equation, which is generally written as [tex]\( y = mx + b \)[/tex]. Here, [tex]\( y \)[/tex] represents the total cost, [tex]\( x \)[/tex] represents the number of toppings, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Components of the Equation:
- [tex]\( b = 12 \)[/tex]: This is the y-intercept, which indicates the starting cost of a medium cheese pizza with no toppings. This means when [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex].
- [tex]\( m = 2 \)[/tex]: This is the slope of the line. The slope tells us how much the total cost [tex]\( y \)[/tex] increases for each additional topping [tex]\( x \)[/tex]. Specifically, for each additional topping, the cost increases by \[tex]$2. 3. Plotting Key Points: To accurately draw the graph, plot key points using the equation: - When \( x = 0 \), \( y = 12 \). This gives us the point (0, 12). - When \( x = 1 \), substitute \( x \) into the equation: \[ y = 12 + 2 \cdot 1 = 14 \] This gives us the point (1, 14). - When \( x = 2 \), substitute \( x \) into the equation: \[ y = 12 + 2 \cdot 2 = 16 \] This gives us the point (2, 16). 4. Slope Interpretation: - The slope of 2 means that for every additional topping (increasing \( x \) by 1), the total cost \( y \) increases by \$[/tex]2. This is a constant rate of increase.
5. Drawing the Graph:
- Start at the y-intercept (0, 12).
- From this point, move right by 1 unit on the x-axis (representing 1 additional topping) and move up by 2 units on the y-axis (representing the increase in cost by \[tex]$2). Plot the next point. - Continue this process to plot several points and draw a straight line through them. 6. Verifying the Correct Graph: - The correct graph will be a straight line starting at (0, 12) and rising with a slope of 2. - Visually, the graph should show a linear relationship where the cost increases by \$[/tex]2 for each additional topping.
By examining the graphs given in the options (e.g., A, B, etc.), select the one that:
- Starts at the point (0, 12) on the y-axis.
- Has a linear rise such that for every additional 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
By following these steps, you can accurately determine which graph represents the equation [tex]\( y = 12 + 2x \)[/tex].
### Step-by-Step Solution:
1. Identify the Equation Format:
The equation [tex]\( y = 12 + 2x \)[/tex] is in the slope-intercept form of a linear equation, which is generally written as [tex]\( y = mx + b \)[/tex]. Here, [tex]\( y \)[/tex] represents the total cost, [tex]\( x \)[/tex] represents the number of toppings, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Components of the Equation:
- [tex]\( b = 12 \)[/tex]: This is the y-intercept, which indicates the starting cost of a medium cheese pizza with no toppings. This means when [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex].
- [tex]\( m = 2 \)[/tex]: This is the slope of the line. The slope tells us how much the total cost [tex]\( y \)[/tex] increases for each additional topping [tex]\( x \)[/tex]. Specifically, for each additional topping, the cost increases by \[tex]$2. 3. Plotting Key Points: To accurately draw the graph, plot key points using the equation: - When \( x = 0 \), \( y = 12 \). This gives us the point (0, 12). - When \( x = 1 \), substitute \( x \) into the equation: \[ y = 12 + 2 \cdot 1 = 14 \] This gives us the point (1, 14). - When \( x = 2 \), substitute \( x \) into the equation: \[ y = 12 + 2 \cdot 2 = 16 \] This gives us the point (2, 16). 4. Slope Interpretation: - The slope of 2 means that for every additional topping (increasing \( x \) by 1), the total cost \( y \) increases by \$[/tex]2. This is a constant rate of increase.
5. Drawing the Graph:
- Start at the y-intercept (0, 12).
- From this point, move right by 1 unit on the x-axis (representing 1 additional topping) and move up by 2 units on the y-axis (representing the increase in cost by \[tex]$2). Plot the next point. - Continue this process to plot several points and draw a straight line through them. 6. Verifying the Correct Graph: - The correct graph will be a straight line starting at (0, 12) and rising with a slope of 2. - Visually, the graph should show a linear relationship where the cost increases by \$[/tex]2 for each additional topping.
By examining the graphs given in the options (e.g., A, B, etc.), select the one that:
- Starts at the point (0, 12) on the y-axis.
- Has a linear rise such that for every additional 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
By following these steps, you can accurately determine which graph represents the equation [tex]\( y = 12 + 2x \)[/tex].
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