Question

compare the table and graph below. which of the functions has a greater slope? function 1 (table with x: 1,2,3,4 and y: -3, -6, -9, -12) function 2 (graph on coordinate grid)

Explanation:

Step1: Find slope of Function 1

Use slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Take points \((1, -3)\) and \((2, -6)\).
\( m_1 = \frac{-6 - (-3)}{2 - 1} = \frac{-3}{1} = -3 \).

Step2: Find slope of Function 2

From graph, points like \((0, 0)\) and \((9, -3)\) (approx).
\( m_2 = \frac{-3 - 0}{9 - 0} = \frac{-3}{9} = -\frac{1}{3} \approx -0.333 \).

Step3: Compare slopes

\( |m_1| = 3 \), \( |m_2| = \frac{1}{3} \). Since \( -3 < -\frac{1}{3} \) in value, but slope "greater" (steeper negative? Or magnitude? Wait, slope magnitude: \( 3 > \frac{1}{3} \). But if "greater" as in value (less negative), Function 2. Wait, no—wait, the problem says "greater slope". Let's check values: \( -3 \) vs \( -\frac{1}{3} \). Since \( -\frac{1}{3} > -3 \) (because it's closer to 0), so Function 2 has greater slope? Wait no, wait Function 1's slope is -3, Function 2's is -1/3. So -1/3 is greater than -3. But wait, maybe I misread Function 1's table. Wait the table: x=1, y=-3; x=2, y=-6; x=3, y=-9; x=4, y=-12. So slope is (y2-y1)/(x2-x1) = (-6 - (-3))/(2-1) = -3. Correct. Function 2: let's take two points. The graph passes through (0,0) and (3, -1)? Wait maybe my initial points were wrong. Wait the x-axis is from 0 to 10, y-axis from -10 to 10? Wait the graph: let's see, the line goes from (0,0) to (9, -3)? Wait no, maybe (0,0) and (3, -1)? Wait no, let's count the grid. Each grid is 1 unit. Let's take (0,0) and (3, -1)? No, maybe (0,0) and (6, -2)? Wait no, the line is decreasing, so slope is negative. Let's recalculate Function 2's slope. Let's take two clear points. Suppose at x=0, y=0; at x=3, y=-1? No, maybe the graph is from (0,0) to (9, -3), so slope is -3/9 = -1/3. Correct. So Function 1: slope -3, Function 2: slope -1/3. So -1/3 is greater than -3 (since it's less negative). So Function 2 has a greater slope. Wait but wait, maybe the problem considers "greater" as in more negative (steeper). But the question is "which of the functions has a greater slope". In math, slope is a number, so greater means larger in value. So -1/3 is greater than -3. So Function 2? Wait no, wait maybe I messed up Function 1's table. Wait the table: x=1, y=-3; x=2, y=-6; so when x increases by 1, y decreases by 3. So slope is -3. Function 2: when x increases by 9, y decreases by 3, so slope -3/9 = -1/3. So -1/3 is greater than -3. So Function 2 has a greater slope.

Answer:

Function 2