Question

use the situation to answer the questions.
a function is used to calculate the side length of a square based upon the area of the square. the side length is calculated by the following formula:
$s = \sqrt{a}$, where $s$ is the side length and $a$ is the area
the formula could also be written using $x$ and $y$:
$y = \sqrt{x}$, where $y$ is the side length and $x$ is the area

1. assuming the area is measured in square meters, complete the table of values.

area ($\text{m}^2$) $x$	process $\sqrt{x}$	side length (m) $y$
1	1	1
2.25	2.25	1.5
4	4	2

1. plot the points and see if they form a straight line.

(graph with x-axis: area of square ($\text{m}^2$), y-axis: side length (m))

Explanation:

For Question 1 (Completing the Table):

Step 1: Recall the formula

The formula for side length \( y \) in terms of area \( x \) is \( y = \sqrt{x} \). We use this to find the side length for each given area.

Step 2: Calculate for each \( x \)

• For \( x = 0.25 \): \( y=\sqrt{0.25} = 0.5 \) (matches the table).
• For \( x = 1 \): \( y=\sqrt{1}=1 \) (matches the table).
• For \( x = 2.25 \): \( y=\sqrt{2.25} = 1.5 \) (matches the table).
• For \( x = 4 \): \( y=\sqrt{4}=2 \) (matches the table).

The table is already correctly filled as the calculations confirm \( y=\sqrt{x} \) gives the side lengths shown.

For Question 2 (Plotting Points and Checking Linearity):

Step 1: Identify the points

The points from the table are \( (0.25, 0.5) \), \( (1, 1) \), \( (2.25, 1.5) \), \( (4, 2) \).

Step 2: Plot the points

• On the graph, the x - axis is "area of square (\( m^2 \))" and the y - axis is "side length (\( m \))".
• Plot \( (0.25, 0.5) \): Move 0.25 units on the x - axis and 0.5 units on the y - axis.
• Plot \( (1, 1) \): Move 1 unit on the x - axis and 1 unit on the y - axis.
• Plot \( (2.25, 1.5) \): Move 2.25 units on the x - axis and 1.5 units on the y - axis.
• Plot \( (4, 2) \): Move 4 units on the x - axis and 2 units on the y - axis.

Step 3: Check for linearity

A linear relationship has the form \( y = mx + b \) (constant slope \( m \)). Let's calculate the slope between consecutive points:

• Between \( (0.25, 0.5) \) and \( (1, 1) \): \( m=\frac{1 - 0.5}{1 - 0.25}=\frac{0.5}{0.75}=\frac{2}{3}\approx0.6667 \)
• Between \( (1, 1) \) and \( (2.25, 1.5) \): \( m=\frac{1.5 - 1}{2.25 - 1}=\frac{0.5}{1.25}=0.4 \)
• Between \( (2.25, 1.5) \) and \( (4, 2) \): \( m=\frac{2 - 1.5}{4 - 2.25}=\frac{0.5}{1.75}=\frac{2}{7}\approx0.2857 \)

Since the slope is not constant, the points do not form a straight line. The graph of \( y = \sqrt{x} \) is a curve (a square - root function graph), not a straight line.

Answer:

(for Question 2):
The points \( (0.25, 0.5) \), \( (1, 1) \), \( (2.25, 1.5) \), \( (4, 2) \) do not form a straight line. The graph of \( y=\sqrt{x} \) is a curve.