In order to shift an equation to the left three units, the x-variable will need to be replaced with the quantity of $\displaystyle (x+3)$.  This shifts all points left three units.

$\displaystyle y=9-\frac{2}{9}(x+3)$

Simplify the equation.

$\displaystyle y=9-\frac{2}{9}x-\frac{2}{3} = \frac{27}{3}-\frac{2}{9}x-\frac{2}{3}$

The answer is:  $\displaystyle y=-\frac{2}{9}x+\frac{25}{3}$

Add six to the equation since a vertical shift will increase the y-intercept by six units.

$\displaystyle y=9(2x-1)+6$

Simplify this equation by distribution.

$\displaystyle y=9(2x)-9(1)+6 = 18x-3$

The answer is:  $\displaystyle y= 18x-3$

Divide by three on both sides.

$\displaystyle \frac{3y}{3}=\frac{4-x}{3}$

The equation becomes:  $\displaystyle y=-\frac{1}{3}x+\frac{4}{3}$

If this equation shifts to the left five units, we will need to replace the x term with the quantity $\displaystyle (x+5)$.

$\displaystyle y=-\frac{1}{3}(x+5)+\frac{4}{3}$

Simplify this equation by distribution.

$\displaystyle y=-\frac{1}{3}(x+5)+\frac{4}{3} = -\frac{1}{3}x-\frac{5}{3}+\frac{4}{3}$

Combine like-terms.

The answer is:  $\displaystyle y= -\frac{1}{3}x-\frac{1}{3}$

Translation of a graph to the left four units will require replacing the x-variable with the quantity:

$\displaystyle (x+4)$

Replace the term inside the equation.

$\displaystyle y=3(x+4)-15$

Use distribution so simplify the terms.

$\displaystyle y=3x+12-15$

Simplify the equation.

The answer is:  $\displaystyle y=3x-3$

In order to find the equation after the translation, we will need to put the equation in slope-intercept format, $\displaystyle y=mx+b$.

Subtract $\displaystyle x$ from both sides of the equation.

$\displaystyle x+3y-x=17-x$

The equation becomes:  $\displaystyle 3y= -x+17$

Divide by three on both sides.

$\displaystyle \frac{3y}{3}= \frac{-x+17}{3}$

$\displaystyle y=-\frac{1}{3}x +\frac{17}{3}$

Add two to the y-intercept for the vertical shift.  This is the same as adding $\displaystyle \frac{6}{3}$.

$\displaystyle y=-\frac{1}{3}x +\frac{17}{3}+\frac{6}{3} = -\frac{1}{3}x+\frac{23}{3}$

The equation is:  $\displaystyle y= -\frac{1}{3}x+\frac{23}{3}$

Rewrite the given equation in standard form to slope intercept format, $\displaystyle y=mx+b$.

Subtract x from both sides.

$\displaystyle x+y-x=-9-x$

The slope intercept form is:  $\displaystyle y=-x-9$

If the line is translated 5 units to the left, we need to replace the quantity of x with $\displaystyle (x+5)$.

$\displaystyle y=-(x+5)-9$

Simplify the equation.  Distribute the negative through the binomial.

$\displaystyle y=-(x+5)-9 = -x-5-9 =-x-14$

The answer is:  $\displaystyle y=-x-14$

Rewrite this equation in slope intercept form $\displaystyle y=mx+b$.

Add $\displaystyle 3x$ on both sides.

$\displaystyle 2y-3x+3x=4+3x$

The equation becomes:

$\displaystyle 2y = 3x+4$

Divide by two on both sides.

$\displaystyle \frac{2y }{2}= \frac{3x+4}{2}$

The equation in slope intercept form is:  $\displaystyle y=\frac{3}{2}x+2$

Shifting this equation down four units means that the y-intercept will be decreased four units.

The answer is:  $\displaystyle y=\frac{3}{2}x-2$

Rewrite the equation $\displaystyle x=3y+1$ in slope-intercept form:   $\displaystyle y=mx+b$

Subtract one from both sides.

$\displaystyle x-1=3y+1-1$

$\displaystyle x-1=3y$

Divide by three on both sides.

$\displaystyle \frac{x-1}{3}=\frac{3y}{3}$

$\displaystyle y=\frac{1}{3}x-\frac{1}{3}$

If this line is shifted to the left three units, replace the x-variable with $\displaystyle (x+3)$.

$\displaystyle y=\frac{1}{3}(x+3)-\frac{1}{3}$

Simplify by distribution.

$\displaystyle y=\frac{1}{3}x+1-\frac{1}{3} = \frac{1}{3}x+\frac{2}{3}$

The answer is:  $\displaystyle y= \frac{1}{3}x+\frac{2}{3}$

If the linear function is shifted left two units, the x-variable must be replaced with the quantity of $\displaystyle (x+2)$.

$\displaystyle y=-\frac{1}{3}(x+2)-1$

Simplify the equation by distribution.

$\displaystyle y=-\frac{1}{3}x-\frac{2}{3}-1= -\frac{1}{3}x-\frac{2}{3}-\frac{3}{3}$

Combine like terms.

The answer is:  $\displaystyle y=-\frac{1}{3}x-\frac{5}{3}$

To shift the line left four units, we will need to replace the x-variable with the quantity of:

$\displaystyle (x+4)$

Replace this term in the original equation.

$\displaystyle y=-3(x+4)-8$

Use distribution to simplify.

$\displaystyle y=-3x-12-8 = -3x-20$

The answer is:  $\displaystyle y= -3x-20$