Question

1. use first and second differences to determine whether each relation is linear, quadratic, or neither. show your work.

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & 20 \\\\ \\hline -1 & 9 \\\\ \\hline 0 & 0 \\\\ \\hline 1 & -7 \\\\ \\hline 2 & -12 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & 39 \\\\ \\hline -1 & 32 \\\\ \\hline 0 & 25 \\\\ \\hline 1 & 18 \\\\ \\hline 2 & 11 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & 48 \\\\ \\hline -1 & 24 \\\\ \\hline 0 & 8 \\\\ \\hline 1 & 0 \\\\ \\hline 2 & 0 \\\\ \\hline \\end{array}$$

1. convert to standard form.

a) \\(y = (x - 7)(x - 2)\\)

Explanation:

Analyze the first table in Question 7

We calculate the first and second differences for the first table:

$$
LATEXBLOCK0
$$

Since the second differences are constant (\(2\)), this relation is quadratic.

Analyze the second table in Question 7

We calculate the first differences for the second table:

$$
LATEXBLOCK1
$$

Since the first differences are constant (\(-7\)), this relation is linear.

Analyze the third table in Question 7

We calculate the first and second differences for the third table:

$$
LATEXBLOCK2
$$

Since the second differences are constant (\(8\)), this relation is quadratic.

Convert the equation in Question 8 to standard form

Using the Polynomial Expansion and Distributive Property concepts, we expand the factored form equation \(y = (x - 7)(x - 2)\):

$$ y = x(x - 2) - 7(x - 2) $$

$$ y = x^2 - 2x - 7x + 14 $$

$$ y = x^2 - 9x + 14 $$

This represents the quadratic relation in standard form.

Answer:

Question 7

First Table

• First differences: \(-11, -9, -7, -5\)
• Second differences: \(2, 2, 2\)
• Classification: Quadratic

Second Table

• First differences: \(-7, -7, -7, -7\)
• Classification: Linear

Third Table

• First differences: \(-24, -16, -8, 0\)
• Second differences: \(8, 8, 8\)
• Classification: Quadratic

---

Question 8

a) The standard form of the equation \(y = (x - 7)(x - 2)\) is:

$$y = x^2 - 9x + 14$$