The Arizona Mathematics Standard you're referencing focuses on helping students develop the skills to represent relationships between quantities using equations and to understand how to accurately graph these equations on coordinate axes. Here's a breakdown of the key concepts and some examples for clarification.

Key Concepts and Clarifications

1. Graphing Equations on Coordinate Axes with Labels and Scales:

   • This means plotting an equation on a coordinate plane (x and y-axis) with proper scaling and labels.
   • Labels: The x-axis and y-axis should be labeled with the quantities they represent, like "Time (seconds)" for x or "Distance (meters)" for y.
   • Scale: Each axis must have a consistent scale, such as counting by 1s, 2s, or 5s, to accurately reflect the relationship between the two quantities.

   Examples for Clarification:

   • Example 1: If you have the equation $y = 2x$, where $x$ represents hours worked and $y$ represents earnings in dollars, label the axes with "Hours Worked" and "Earnings in Dollars."
   • Example 2: For an equation like $y = -3x + 5$, label and scale the axes so that increments of 1 can show the intercepts and slope.
   • Example 3: Use the equation $y = x^2$ to demonstrate scaling on both axes, choosing an appropriate scale to fit the curve.
   • Example 4: Graphing $y = \frac{1}{2}x + 3$ requires selecting a scale that makes plotting fractional slopes easier, such as marking intervals of 0.5.

2. Understanding Relationships Between Quantities:

   • This involves recognizing how two variables are connected. For example, as one variable increases, the other might also increase (positive relationship) or decrease (negative relationship).
   • Understanding this relationship helps interpret real-world scenarios, like distance over time or cost based on quantity.

   Examples for Clarification:

   • Example 1: A linear relationship, such as $y = 3x + 2$, where $x$ is the number of products sold, and $y$ is the revenue.
   • Example 2: In a quadratic relationship, $y = x^2$, as the value of $x$ changes, $y$ changes at a faster rate, representing acceleration or areas.
   • Example 3: In a proportional relationship like $y = 5x$, doubling $x$ doubles $y$, representing constant growth, such as hourly wage.
   • Example 4: A decreasing relationship, $y = -2x + 8$, where increasing $x$ reduces $y$, representing things like cost reduction as production increases.

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10 Practice Questions for Worksheet

1. Write the equation of a line in slope-intercept form given that the line has a slope of 4 and passes through the origin. Graph this equation on a coordinate plane with labels and a scale.

2. Create an equation that represents the relationship between distance (y) in miles and time (x) in hours, where the speed is constant at 60 miles per hour. Graph the equation with appropriate labels and scales.

3. You earn $12 per hour for tutoring. Write an equation that represents your total earnings (y) based on hours worked (x), and then graph the equation on a coordinate plane.

4. A company charges a fixed base fee of $50, plus $20 per hour of service. Write an equation representing the total cost (y) based on hours of service (x). Graph this equation with appropriate labels and scales.

5. Graph the equation $y = -2x + 5$ and identify the slope and y-intercept. Label the axes and choose a scale that fits the line on a standard coordinate plane.

6. **Write an equation for the relationship where y is three times x. Graph this equation with appropriate labels and scales, showing at least three