Understand Linear and Nonlinear Functions: CCSS.Math.Content.8.F.A.3 - 8th Grade Math

Start by visualizing the graph associated with the function :

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 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 :

We have the following answer choices.

1. 
2. 
3. 
4. 

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.

Starting with

moves the parabola by units to the right.

Similarly moves the parabola by units to the left.

Hence the correct answer is option .

A function may only have one y-value for each x-value.

The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.

The highest exponent of the variable term is two (). This tells that this function is quadratic, meaning that it is a parabola.

The graph below will be the answer, as it shows a parabolic curve.

From the vertex, we know that the equation of the parabola will take the form for some .

To calculate that , we plug in the values from the other point we are given, , and solve for :

Now the equation is . This is not an answer choice, so we need to rewrite it in some way.

Expand the squared term:

Distribute the fraction through the parentheses:

Combine like terms:

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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

Next, we can divide each side by

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 to both sides:

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 to both sides:

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

Next, we can divide each side by

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

Next, we can divide each side by

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

For this equation, we can solve for 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 from both sides:

Next, we can divide each side by

This equation is in slope-intercept form; thus, 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:

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:

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

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

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

This equation is in slope-intercept form; thus, is the correct answer.

Start by visualizing the graph associated with the function :

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 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 :

We have the following answer choices.

1. 
2. 
3. 
4. 

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.