Question

complete the function table for each equation.
y = -6x

1. y = x - 2
2. y = -\frac{1}{5}x - 5
3. y = 4x + 9
4. y = \frac{1}{3}x + 2
5. y = x - 9
6. y = 6x + 3
7. y = -8x
8. y = -9x

y = 7x - 8

1. y = -\frac{1}{3}x + 4
2. y = x - 7

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Explanation:

Let's take problem 2: \( y = 4x + 9 \) as an example to show the step - by - step solution for completing the function table.

Step 1: When \( x = 6 \)

Substitute \( x = 6 \) into the equation \( y=4x + 9 \).
According to the substitution rule, we replace \( x \) with 6 in the expression \( 4x+9 \).
So \( y = 4\times6+9 \)
First, calculate the multiplication: \( 4\times6 = 24 \)
Then, calculate the addition: \( 24 + 9=33 \)

Step 2: When \( x=-3 \)

Substitute \( x = - 3 \) into \( y = 4x+9 \)
\( y=4\times(-3)+9 \)
First, calculate the multiplication: \( 4\times(-3)=-12 \)
Then, calculate the addition: \( - 12 + 9=-3 \)

Step 3: When \( x = 9 \)

Substitute \( x = 9 \) into \( y = 4x+9 \)
\( y = 4\times9+9 \)
First, calculate the multiplication: \( 4\times9 = 36 \)
Then, calculate the addition: \( 36+9 = 45 \)

Step 4: When \( x = 3 \)

Substitute \( x = 3 \) into \( y = 4x+9 \)
\( y=4\times3+9 \)
First, calculate the multiplication: \( 4\times3 = 12 \)
Then, calculate the addition: \( 12 + 9=21 \)

Step 5: When \( x=-2 \)

Substitute \( x=-2 \) into \( y = 4x+9 \)
\( y=4\times(-2)+9 \)
First, calculate the multiplication: \( 4\times(-2)=-8 \)
Then, calculate the addition: \( - 8+9 = 1 \)

We can follow the same process for other problems. For example, for problem 3: \( y=6x + 3 \)

• When \( x=-7 \): \( y=6\times(-7)+3=-42 + 3=-39 \)
• When \( x=-6 \): \( y=6\times(-6)+3=-36 + 3=-33 \)
• When \( x=-1 \): \( y=6\times(-1)+3=-6 + 3=-3 \)
• When \( x = 1 \): \( y=6\times1+3=6 + 3=9 \)
• When \( x = 7 \): \( y=6\times7+3=42 + 3=45 \)

For problem 6: \( y=\frac{1}{3}x + 2 \)

• When \( x = 4 \): \( y=\frac{1}{3}\times4+2=\frac{4}{3}+2=\frac{4 + 6}{3}=\frac{10}{3}\approx3.33 \)
• When \( x = 7 \): \( y=\frac{1}{3}\times7+2=\frac{7}{3}+2=\frac{7 + 6}{3}=\frac{13}{3}\approx4.33 \)
• When \( x=-1 \): \( y=\frac{1}{3}\times(-1)+2=-\frac{1}{3}+2=\frac{-1 + 6}{3}=\frac{5}{3}\approx1.67 \)
• When \( x=-7 \): \( y=\frac{1}{3}\times(-7)+2=-\frac{7}{3}+2=\frac{-7 + 6}{3}=-\frac{1}{3}\approx - 0.33 \)
• When \( x = 5 \): \( y=\frac{1}{3}\times5+2=\frac{5}{3}+2=\frac{5+6}{3}=\frac{11}{3}\approx3.67 \)

(We can continue this process for all the given functions in the table. Since the problem is to complete the function tables, we can use the substitution method for each \( x \) value in the table to find the corresponding \( y \) value as shown above.)

Answer:

Taking problem 2 (\( y = 4x + 9 \)) as an example, the completed table (partial) is:

\( x \)	\( y \)
- 3	- 3
9	45
3	21
- 2	1

(For other problems, we can use the same substitution method to find the \( y \) values for each given \( x \) value in their respective tables.)