Question

a square that has a side length of 9 cm has an area of 81 cm². the relationship between the side length and the area of the square is a function.
a. complete the table with the area for each given side length.

side length (cm)	area (cm²)
2
4
6
s

then, write a rule for a function, a, that gives the area of the square in cm² when the side length is s cm. use function notation.

b. what does a(2) represent in this situation? what is its value?

c. on the coordinate plane, sketch a graph of this function.

(coordinate plane with y - axis labeled area (cm²) from 0 to 50 and x - axis labeled side length (cm) from 0 to 8)

Explanation:

Part (a)

Step 1: Recall the formula for the area of a square

The area \( A \) of a square with side length \( s \) is given by \( A = s^2 \).

Step 2: Calculate the area for each side length

• For \( s = 1 \): \( A = 1^2 = 1 \)
• For \( s = 2 \): \( A = 2^2 = 4 \)
• For \( s = 4 \): \( A = 4^2 = 16 \)
• For \( s = 6 \): \( A = 6^2 = 36 \)
• For a general side length \( s \): \( A(s) = s^2 \)

Filling the table:

Side Length (cm)	Area (cm²)
2	4
4	16
6	36
\( s \)	\( s^2 \)

The function rule is \( A(s) = s^2 \).

Part (b)

• \( A(2) \) represents the area of the square when the side length is \( 2 \) cm.
• To find its value, we use the function \( A(s) = s^2 \). Substitute \( s = 2 \): \( A(2) = 2^2 = 4 \).

Answer:

\( A(2) \) represents the area of the square with side length \( 2 \) cm, and its value is \( 4 \) \( \text{cm}^2 \).

Part (c)

To sketch the graph of \( A(s) = s^2 \) (where \( s \geq 0 \) since side length can't be negative):

1. Identify key points: Use the values from the table: \( (1, 1) \), \( (2, 4) \), \( (4, 16) \), \( (6, 36) \).
2. Plot the points: On the coordinate plane, plot these points where the x - axis is the side length \( s \) (in cm) and the y - axis is the area \( A \) (in \( \text{cm}^2 \)).
3. Draw the curve: Since \( A(s)=s^2 \) is a quadratic function, the graph is a parabola opening upwards. Connect the plotted points with a smooth curve. For example, when \( s = 3 \), \( A(3)=9 \), so the point \( (3,9) \) is also on the graph, and we can include such additional points for a more accurate sketch.