# Calculus 1 : How to graph functions of points

## Example Questions

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### Example Question #1 : How To Graph Functions Of Points

Find the critical points (rounded to two decimal places):

Explanation:

To find the critical points, set  and solve for .

Differentiate:

Set equal to zero:

Solve for  using the quadratic formula:

### Example Question #2 : How To Graph Functions Of Points

Find the  value(s) of the critical point(s) of

.

There are no real answers.

Explanation:

In order to find the critical points, we must find  and solve for

Set

Use the quadratic equation to solve for .

Remember that the quadratic equation is as follows.

, where a,b and c refer to the coefficents in the

equation  .

In this case, , , and .

After plugging in those values, we get

.

So the critical points  values are:

### Example Question #3 : How To Graph Functions Of Points

Find the  value(s) of the critical point(s) of

.

Explanation:

In order to find the critical points, we must find  and solve for .

Set

Use the quadratic equation to solve for .

Remember that the quadratic equation is as follows.

, where a,b and c refer to the coefficents in the equation  .

In this case, , , and .

After plugging in those values, we get.

So the critical points  values are,

### Example Question #4 : How To Graph Functions Of Points

Find the critical points of

The critical points are complex.

Explanation:

First we need to find .

Now we set

Now we can use the quadratic equation in order to find the critical points.

Remember that the quadratic equation is

,

where a,b,c refer to the coefficients in the equation

In this case, a=3, b=6, and c=1.

Thus are critical points are

### Example Question #1 : Other Points

Find the critical points of

.

There are no critical points.

Explanation:

In order to find the critical points, we need to find  using the power rule .

Now we set , and solve for .

Thus  is a critical point.

### Example Question #6 : How To Graph Functions Of Points

Find the critical point(s) of .

and

and

and

Explanation:

To find the critical point(s) of a function , take its derivative , set it equal to , and solve for .

Given , use the power rule

to find the derivative. Thus the derivative is, .

Since :

The critical point  is

### Example Question #7 : How To Graph Functions Of Points

Find the critical points of

.

There are no critical points

Explanation:

In order to find the critical points, we must find  using the power rule .

.

Now we set .

Now we use the quadratic equation in order to solve for .

Remember that the quadratic equation is as follows.

,

where a,b,c correspond to the coefficients in the equation

.

In this case, a=9, b=-40, c=4.

Then are critical points are:

### Example Question #1 : How To Graph Functions Of Points

Find all the critical points of

.

There are no critical points.

Explanation:

In order to find the critical points, we first need to find  using the power rule ..

Now we set .

Thus the critical points are at

, and

.

### Example Question #9 : How To Graph Functions Of Points

Find the critical points of the following function:

Explanation:

To find critical points the derivative of the function must be found.

Critical points occur where the derivative equals zero.

### Example Question #10 : How To Graph Functions Of Points

Determine the point on the graph that is not changing if .

Explanation:

To find the point where the graph of  is not changing, we must set the first derivative equal to zero and solve for .

To evaluate this derivate, we need the following formulae:

Now, setting the derivate equal to  to find where the graph is not changing:

Now, to find the corresponding  value, we plug this  value back into :

Therefore, the point where  is not changing is

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