Question

function 1:
function 2:
x | 1 | 2 | 3 | 4 | 5
p(x) | 0 | 3 | 13 | 16 | 19

Explanation:

Step1: Find slope of Function 1 (q(x))

From the graph, q(x) passes through (0, 5) and (3, 0). Slope \( m_1 = \frac{0 - 5}{3 - 0} = -\frac{5}{3} \approx -1.67 \).

Step2: Find slope of Function 2 (p(x))

Using points (1, 0), (2, 3): slope \( m_2 = \frac{3 - 0}{2 - 1} = 3 \). (Check other points: (3,13) - slope from (2,3) to (3,13) is 10, so p(x) is not linear. Wait, maybe initial points: Wait, x=1, p(x)=0; x=2, p(x)=3; x=3, p(x)=13. The differences: 3-0=3, 13-3=10, not constant. But Function 1 is linear. Wait, maybe the question is to compare rates? Wait, maybe find the rate of change (slope) for Function 1 (linear) and average rate for Function 2. But assuming we need to find which has greater rate. Wait, Function 1: slope is -5/3 ≈ -1.67 (negative, decreasing). Function 2: from x=1 to x=5, change in p(x) is 19-0=19, change in x=4, average rate 19/4=4.75. But maybe the question is about the slope (rate of change) of the linear function (Function 1) and the rate of Function 2. Wait, maybe the problem is to find which function has a greater rate of change (slope for linear, average for non-linear). But since Function 1 is linear with slope -5/3, and Function 2's average rate from x=1 to x=5 is 19/4=4.75, which is greater than |-5/3|≈1.67. But maybe the original problem was to compare the rates. Alternatively, maybe find the equation of Function 1. Let's re-express:

Function 1: passes through (0, 5) and (3, 0). So equation: \( y = -\frac{5}{3}x + 5 \).

Function 2: let's check if it's linear? No, since 3-0=3, 13-3=10, 16-13=3, 19-16=3. Wait, x=3 to 4: 16-13=3, x=4 to 5: 19-16=3. Oh, maybe I misread: x=1:0, x=2:3, x=3:13? Wait, no, maybe the table is x:1,2,3,4,5 and p(x):0,3,13,16,19? Wait, 3 to 13 is +10, 13 to 16 is +3, 16 to 19 is +3. So not linear. But Function 1 is linear. Maybe the question is to find the rate of change (slope) of the linear function (Function 1) and the average rate for a part of Function 2.

Assuming the problem is to find the slope of Function 1:

Using two points (0, 5) and (3, 0):

Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 5}{3 - 0} = -\frac{5}{3} \).

Equation of Function 1: \( q(x) = -\frac{5}{3}x + 5 \).

For Function 2, if we take x=1 to x=2: rate is 3, x=2 to x=3: 10, x=3 to x=4: 3, x=4 to x=5: 3.

But maybe the original problem was to compare the rates. Since the user's image shows two functions, likely the task is to find the rate of change (slope) for the linear function (Function 1) and analyze Function 2.

Answer:

Assuming the question is to find the slope of Function 1 (q(x)): The slope is \( \boldsymbol{-\frac{5}{3}} \) (or approximately -1.67). If comparing rates, Function 2 has a greater average rate from x=1 to x=5 (19/4 = 4.75) than Function 1's slope magnitude (5/3 ≈ 1.67).