Question

the graphs below shows some properties of regular polygons.
when compared with the independent variable, how many of the graphs represent a linear relationship?
options: 0, 1, 2, 3
(there is a graph with x - axis as number of sides and y - axis as # of diagonals from 1 vertex, and another graph partially shown.)

Explanation:

Step1: Recall linear relationship

A linear relationship has a constant rate of change (slope), meaning the graph is a straight line (or points lie on a straight line with constant slope).

Step2: Analyze the first graph (diagonals vs sides)

For a regular polygon, the number of diagonals from one vertex \( d \) and number of sides \( n \) has the formula \( d=n - 3 \) (for \( n\geq3 \)). Let's check the points:

• When \( n = 3 \) (triangle), \( d=0 \) (matches the point (3,0)).
• When \( n = 4 \) (quadrilateral), \( d = 4-3=1 \) (matches (4,1)).
• When \( n = 5 \) (pentagon), \( d=5 - 3 = 2 \) (matches (5,2)).
• When \( n = 6 \) (hexagon), \( d=6 - 3=3 \) (matches (6,3)).

The slope between consecutive points: from (3,0) to (4,1): slope \( m=\frac{1 - 0}{4 - 3}=1 \); from (4,1) to (5,2): \( m=\frac{2 - 1}{5 - 4}=1 \); from (5,2) to (6,3): \( m=\frac{3 - 2}{6 - 5}=1 \). So this is a linear relationship (slope = 1, equation \( d=n - 3 \)).

Step3: Analyze the second graph (partially shown, but assume we check linearity)

Wait, but the first graph (diagonals) is linear. Wait, but maybe the other graph (the one with y=-20? Wait, the second graph's y-axis starts at -20, but without full points, but the first graph is linear. Wait, no—wait, the question is about "the graphs" (plural). But maybe the second graph (if it's, say, interior angles or something else). Wait, no, the first graph (diagonals) is linear. Wait, but let's re - check. Wait, the formula for diagonals from one vertex is linear (\( d=n - 3 \)), so that's one linear graph. But maybe the other graph (the one with y=-20) — but since we can only see part, but the first graph is linear. Wait, no, wait the options are 0,1,2,3. Wait, maybe I made a mistake. Wait, the first graph: points (3,0), (4,1), (5,2), (6,3). The slope between (3,0) and (4,1) is 1, (4,1) and (5,2) is 1, (5,2) and (6,3) is 1. So it's a linear relationship (straight line with slope 1). Now, what about other graphs? The second graph is partially shown, but maybe it's about another property. Wait, but the question is "how many of the graphs" — but in the given image, there are two graphs? Wait, the first is diagonals vs sides, the second is some other property. Wait, but maybe the second graph (if it's, say, sum of interior angles: sum \( S=(n - 2)\times180 \), which is also linear (slope 180). But the first graph (diagonals from one vertex) is linear (\( d=n - 3 \)), so that's one. Wait, no, wait the first graph's points: (3,0), (4,1), (5,2), (6,3). The line is \( y=x - 3 \), which is linear. So that's one linear graph. Wait, but maybe the second graph (the one with y=-20) — if it's, say, the number of sides and something else, but since we can't see all points, but the first graph is linear. Wait, but the options include 1. Wait, maybe I misread. Wait, the first graph: the x - axis is number of sides (3,4,5,6), y - axis is number of diagonals from one vertex. The relationship \( d=n - 3 \) is linear, so that's one linear graph. So the number of linear graphs is 1? Wait, no—wait, the formula \( d=n - 3 \) is linear, so the graph of \( d \) vs \( n \) is a straight line. So that's one linear graph. So the answer should be 1.

Answer:

1 (corresponding to the option "1")