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:  $\displaystyle y= -3x-20$

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

$\displaystyle y=-9x+9$

Shifting this equation up one unit will add one to the y-intercept.

$\displaystyle y=-9x+9+(1) = -9x+10$

The equation becomes:  $\displaystyle y= -9x+10$

To shift this equation left three units, we will need to replace the x-variable with the quantity $\displaystyle (x+3)$.

$\displaystyle y= -9(x+3)+10$

Use distribution to simplify the equation.

$\displaystyle y= -9x-27+10 = -9x-17$

The answer is:  $\displaystyle y= -9x-17$

If the line is shifted right 5 units, we will need to replace the x-variable of the equation with the quantity $\displaystyle (x-5)$.

$\displaystyle y=3(x-5)-29$

Simplify by distribution.

$\displaystyle y=3x-15-29 = 3x-44$

The answer is:  $\displaystyle y= 3x-44$

Before we apply any transformations, we will need to put the equation in slope-intercept form, $\displaystyle y=mx+b$.

Subtract $\displaystyle x$ from both sides.

$\displaystyle x+9y-x=3x+2-x$

$\displaystyle 9y=2x+2$

Divide by nine on both sides.

$\displaystyle \frac{9y}{9}=\frac{2x+2}{9}$

$\displaystyle y=\frac{2}{9}x+\frac{2}{9}$

Shifting the equation up two units will add 2, or $\displaystyle \frac{18}{9}$ to the y-intercept.

$\displaystyle y=\frac{2}{9}x+\frac{2}{9}+\frac{18}{9}$

The answer is:  $\displaystyle y=\frac{2}{9}x+\frac{20}{9}$

To shift the line left 8 units, we will need to replace the x-value with $\displaystyle (x+8)$.

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

Distribute the negative four through the binomial.

$\displaystyle y=-4x-32 -8$

Combine like-terms.  

The answer is:  $\displaystyle y=-4x-40$

To shift the equation left eight units, replace the x-variable with the quantity of $\displaystyle (x+8)$.

$\displaystyle y=-9(x+8)-5$

Simplify this equation.

$\displaystyle y=-9x-72-5$

Combine like-terms.

The answer is:  $\displaystyle y=-9x-77$

Rewrite the given line in standard form to slope-intercept format:

$\displaystyle y=mx+b$

Subtract $\displaystyle x$ on both sides.

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

$\displaystyle y=-x+9$

Replace the x-value with the quantity of $\displaystyle (x+9)$ since we are shifting the equation left 9 units.

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

Distribute the negative sign through both terms of the binomial.

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

The answer is:  $\displaystyle y=-x$