Question

1. which of the following equations could represent the relationship between x and y in the table below?

x | 1 | 2 | 3 | 4
y | 3 | 9 | 19 | 33
\\( y = 2x^2 + 1 \\)
\\( y = 3x \\)
\\( y = x^2 + 2 \\)
\\( y = \frac{1}{3}x \\)

Explanation:

Step1: Test \( x = 1 \)

Substitute \( x = 1 \) into each equation:

• For \( y = 2x^2 + 1 \): \( y = 2(1)^2 + 1 = 3 \) (matches \( y = 3 \) in the table).
• For \( y = 3x \): \( y = 3(1) = 3 \) (matches, but check other values).
• For \( y = x^2 + 2 \): \( y = 1^2 + 2 = 3 \) (matches, check other values).
• For \( y = \frac{1}{3}x \): \( y = \frac{1}{3}(1) \approx 0.33 \) (does not match).

Step2: Test \( x = 2 \)

Substitute \( x = 2 \):

• For \( y = 2x^2 + 1 \): \( y = 2(4) + 1 = 9 \) (matches \( y = 9 \)).
• For \( y = 3x \): \( y = 3(2) = 6 \) (does not match \( y = 9 \)).
• For \( y = x^2 + 2 \): \( y = 4 + 2 = 6 \) (does not match \( y = 9 \)).

Step3: Confirm with \( x = 3 \) and \( x = 4 \)

• For \( x = 3 \), \( y = 2(3)^2 + 1 = 19 \) (matches table \( y = 19 \)).
• For \( x = 4 \), \( y = 2(4)^2 + 1 = 33 \) (matches table \( y = 33 \)).

Thus, the equation \( y = 2x^2 + 1 \) fits all values.

Answer:

A. \( y = 2x^2 + 1 \)