Question

aiden approximates the area of circles using the equation $a = 3r^2$ and records areas of circles with different radius lengths in a table. complete the following.

radius (in.)	3	4	5	6	7
area (in²)	27	48	75	108	147

a. graph the ordered pairs from the table. choose the correct graph below.
click here to view graph c.
click here to view graph d.
click here to view graph a.
click here to view graph b.

b. is the relation a function? explain.
the relation │ is │ a function because │ each x - value │ corresponds │ to exactly │ one y - value.

c. is the relation a linear function? explain.
the relation │ a linear function because │

Explanation:

Part b

To determine if a relation is a function, we use the definition: a relation is a function if each input (x - value, here radius) has exactly one output (y - value, here area). In the given table, each radius (3, 4, 5, 6, 7) has a unique area. So no x - value is repeated, and each x - value maps to exactly one y - value.

A linear function has a constant rate of change (slope) and its equation is in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept). For a linear function, the difference in \(y\) - values ( \(\Delta y\)) for a constant difference in \(x\) - values ( \(\Delta x\)) should be constant. Let's check the rate of change. For \(x_1 = 3,y_1 = 27\); \(x_2 = 4,y_2 = 48\); the rate of change \(m_1=\frac{48 - 27}{4 - 3}=\frac{21}{1}=21\). For \(x_2 = 4,y_2 = 48\); \(x_3 = 5,y_3 = 75\); the rate of change \(m_2=\frac{75 - 48}{5 - 4}=\frac{27}{1}=27\). Since \(m_1
eq m_2\), the rate of change is not constant. Also, the equation \(A = 3r^{2}\) is a quadratic equation (degree 2), not linear (degree 1). So the relation is not a linear function because the rate of change is not constant (or the equation is quadratic, not in the form of a linear function).

Answer:

The relation is a function because each x - value (radius) corresponds to exactly one y - value (area).

Part c