To analyze the graph with the given information, letβs go step by step.
A. Identify the \(y\)-intercept and the slope of the graph.
For a linear equation in the form \(y = mx + b\):
- \(y\)-intercept (b): This is where the graph crosses the y-axis. For example, if the graph crosses at (0, 1), then the \(y\)-intercept is \(1\).
- Slope (m): This is the change in \(y\) over the change in \(x\). If moving from point (0, 1) to point (2, -2) involves a drop of 3 units in \(y\) (from 1 to -2) and a rightward movement of 2 units in \(x\), the slope \(m\) is calculated as: \[ m = \frac{\Delta y}{\Delta x} = \frac{-2 - 1}{2 - 0} = \frac{-3}{2} = -\frac{3}{2} \] Thus, the slope is \(-\frac{3}{2}\), and the \(y\)-intercept is \(1\).
B. Write the linear equation that represents the graph.
With the identified \(y\)-intercept and slope, we can write the equation: \[ y = -\frac{3}{2}x + 1 \]
C. Explain how you can use the \(y\)-intercept and slope of a graph to write a linear equation that represents the graph.
To write a linear equation from the graph, we identify two critical components:
- The \(y\)-intercept tells us the value of \(y\) when \(x = 0\). This gives us the \(b\) term in the slope-intercept form \(y = mx + b\).
- The slope indicates how much \(y\) changes for a unit change in \(x\). It is represented as \(m\) in the equation. The slope can be calculated by choosing any two distinct points on the line, calculating the rise over the run (change in \(y\) over change in \(x\)).
Combining both gives us the complete linear equation.
D. Create a real-world situation that can go with the equation you wrote in part B and the graph.
Real-World Situation: Consider a toy company that produces and sells custom toy cars. The company has a fixed cost of \(1\) dollar for basic materials needed for production, which represents the \(y\)-intercept. For every toy car produced, they lose \(1.50\) dollars due to operational costs (represented as a negative slope of \(-\frac{3}{2}\)). Therefore, if \(x\) is the number of toy cars produced, the equation \(y = -\frac{3}{2}x + 1\) represents their profit \(y\).
E. Use the graph and the real-world situation you created in part D to make a prediction.
Using the equation \(y = -\frac{3}{2}x + 1\):
- If the company chooses to produce \(0\) toy cars, their profit is: \[ y = -\frac{3}{2}(0) + 1 = 1 \quad (\text{profit of } $1) \]
- If they decide to produce \(2\) toy cars: \[ y = -\frac{3}{2}(2) + 1 = -3 + 1 = -2 \quad (\text{loss of } $2) \]
- If they produce \(1\) toy car: \[ y = -\frac{3}{2}(1) + 1 = -\frac{3}{2} + 1 = -\frac{1}{2} \quad (\text{loss of } $0.5) \]
Prediction: Based on the equation, if the company continues to produce toy cars at this operating cost, they will incur a loss with each additional toy produced beyond what would be sustainable (considering their operational costs). For profitable operation, they need to either decrease operational costs, increase pricing, or reduce production.
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