In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle y=2x^3+9$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle 3x^2+8=y+4$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 5x^2=y+19$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

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

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

We can add $\displaystyle 3x$ to both sides:

$\displaystyle \frac{\begin{array}[b]{r}y-3x=4x+3\\ +3x+3x\ \ \ \ \ \end{array}}{\\\\y=7x+3}$

This equation is in slope-intercept form; thus, $\displaystyle y-3x=4x+3$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle 8y=-3x^3-2$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle -x^2+4=2y-21$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle -8x^2=y-8$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 2y+4x=8x+20$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 4x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}2y+4x=8x+20\\ -4x-4x\ \ \ \ \ \ \end{array}}{\\\\2y=4x+20}$

Next, we can divide each side by $\displaystyle 2\textup:$

$\displaystyle \frac{2y}{2}=\frac{4x+20}{2}=2x+10$

$\displaystyle y=2x+10$

This equation is in slope-intercept form; thus, $\displaystyle 2y+4x=8x+20$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle y=-3x^3-2$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle 3x^2+4=-2x^2+y-15$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 3x^2=2y+2$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle y+9x=3x+17$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 9x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}y+9x=3x+17\\ -9x-9x\ \ \ \ \ \ \ \end{array}}{\\\\y=-6x+17}$

This equation is in slope-intercept form; thus, $\displaystyle y+9x=3x+17$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle 7=x^3-2-3y$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle x^2+4=7x^2+2y-15$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle x^2=2y+24$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 3y+12x=-6x+9$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 12x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}3y+12x=-6x+9\\ -12x-12x\ \ \ \ \ \end{array}}{\\\\3y=-18x+9}$

Next, we can divide each side by $\displaystyle 3\textup:$

$\displaystyle \frac{3y}{3}=\frac{-18x+9}{3}=-6x+3$

$\displaystyle y=-6x+3$

This equation is in slope-intercept form; thus, $\displaystyle 3y+12x=-6x+9$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle 2y=x^3-8$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle 4x^2+9=6y$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 3x^2=y+7$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle y+8x=24$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 8x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}y+8x=24\\ -8x-8x\end{array}}{\\\\y=-8x+24}$

This equation is in slope-intercept form; thus, $\displaystyle y+8x=24$ is the correct answer.

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle 6y=x^3-18$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle x^2+10=3y-20$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 12x^2=2y+8$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle y=2x+8$

This equation is in slope-intercept form; thus, $\displaystyle y=2x+8$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle y=9x^3-27$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle 4x^2+8=y-12$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 3x^2=y+26$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle y+6x=2x-14$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 6x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}y+6x=2x-14\\ -6x-6x\ \ \ \ \ \ \ \end{array}}{\\\\y=-4x+14}$

This equation is in slope-intercept form; thus, $\displaystyle y+6x=2x-14$ is the correct answer. 

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle y=3x^3+21$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle -6x^2-12=y+24$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle -4x^2=-y-6$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle y=-x+1$

This equation is in slope-intercept form; thus, $\displaystyle y=-x+1$ is the correct answer. 

Start by visualizing the graph associated with the function $\displaystyle y=x^2$:

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of $\displaystyle y=x^2-2$ looks like this:

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function $\displaystyle y=(x-1)^2-2$ :

In order to determine if an equation defines a linear function, we want to make sure that the equation of the line is in slope-intercept form:

$\displaystyle y=mx+b$

If we are unable to put an equation in this form, then the equation is not linear. 

Let's take a look at our answer choices:

$\displaystyle 3y=9x^3-36$ 

Notice that in this equation our $\displaystyle x$ value is to the third power, which does not match our slope-intercept form. 

$\displaystyle x^2-8=y+21$

Though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 5x^2=25y+10$

Again, though this equation is not written in $\displaystyle y=$ form, we can tell straight away that this does not define a linear function because the $\displaystyle x$ value is to the second power. 

$\displaystyle 3y+12x=21$

For this equation, we can solve for $\displaystyle y$ to make sure this equation can be written is slope-intercept form. From first glance it looks to be correct because none of our variables are written to a power. In order to tell for certain, we need to isolate the y variable on the left side of the equation.

First, we can subtract $\displaystyle 12x$ from both sides:

$\displaystyle \frac{\begin{array}[b]{r}3y+12x=21\\ -12x-12x\end{array}}{\\\\3y=-12x+21}$

Next, we can divide each side by $\displaystyle 3\textup:$

$\displaystyle \frac{3y}{3}=\frac{-12x+21}{3}=-4x+7$

$\displaystyle y=-4x+7$

This equation is in slope-intercept form; thus, $\displaystyle 3y+12x=21$ is the correct answer.