Question

finding the slope of a line continued
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11 compare finding the slope using a table and using a graph.

Explanation:

For Question 11 (Compare finding the slope using a table and using a graph)

• Using a Table: To find the slope from a table, we identify two ordered pairs \((x_1,y_1)\) and \((x_2,y_2)\) from the table. Then we use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). We calculate the difference in \(y\)-values (vertical change) and the difference in \(x\)-values (horizontal change) between the two points and divide them.
• Using a Graph: For a graph, we can either:
• Identify two points on the line (e.g., where the line crosses grid intersections for easier calculation). Then use the slope formula \(m = \frac{\text{rise}}{\text{run}}\), where "rise" is the vertical change (change in \(y\)) and "run" is the horizontal change (change in \(x\)) between the two points.
• Alternatively, we can visually determine the rate of change by seeing how much \(y\) changes for a given change in \(x\) (e.g., if for every 1 unit increase in \(x\), \(y\) increases by 2 units, the slope is 2).

• Similarities: Both methods rely on the slope formula \(m=\frac{\text{change in }y}{\text{change in }x}\) (or \(m=\frac{y_2 - y_1}{x_2 - x_1}\)). They both require identifying two points (either from the table or the graph) to calculate the rate of change between them.
• Differences:
• Source of Points: In a table, the points are given as ordered pairs (we just pick two rows). In a graph, we have to visually locate points (which might be easier or harder depending on the graph's clarity and if it passes through grid intersections).
• Visual vs. Numerical: Using a graph can be more intuitive for understanding the "steepness" visually (through rise over run), while using a table is more about numerical differences between given values.
• Accuracy: If the graph is not drawn to scale or the points don't lie on grid lines, calculating slope from a graph might be less accurate than from a table (where values are exact). If the graph is clear and points are on grid lines, it can be very straightforward.

Step 1: Identify two points on the line

From the graph, let's take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(5, 100)\) (assuming when \(x = 5\) seasons, \(y = 100\) games).

Step 2: Apply the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 5\), \(y_2 = 100\).

\(m=\frac{100 - 0}{5 - 0}=\frac{100}{5}=20\)

Step 1: Identify two points

Take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(10, 60)\).

Step 2: Use slope formula

\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{60 - 0}{10 - 0}=\frac{60}{10}=6\)

Answer:

• Using a Table: Select two ordered pairs \((x_1,y_1)\), \((x_2,y_2)\) from the table. Use \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• Using a Graph: Identify two points on the line. Use \(m=\frac{\text{rise}}{\text{run}}\) (vertical change over horizontal change between the points).
• Similarities: Both use \(m=\frac{\Delta y}{\Delta x}\) (change in \(y\) over change in \(x\)) and require two points.
• Differences: Table uses given ordered pairs; graph uses visually identified points (easier to see steepness, but accuracy depends on graph clarity).

For Finding Slope of the Lines (assuming we take one example, say Problem 8: "Games vs. Seasons" graph)

Let's assume the graph for Problem 8 has points like \((0,0)\) and \((5, 100)\) (assuming the y-axis is "Games" with values 0,40,80,120... and x-axis "Seasons" 0 - 6).