Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Hyperbolas

Using the information below, determine the equation of the hyperbola.

Foci:  and 

Eccentricity: 

Possible Answers:

Correct answer:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci = 

Distance between vertices = 

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 12, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of 

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 4, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 12, we can conclude that 

Center point: 

Thus, the equation of the hyperbola is:

Example Question #11 : Pre Calculus

Using the information below, determine the equation of the hyperbola.

Foci:  and 

Eccentricity: 

Possible Answers:

Correct answer:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci = 

Distance between vertices = 

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 8, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of 

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 8, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 8, we can conclude that 

Center point: 

Thus, the equation of the hyperbola is:

Example Question #1 : Derivatives

The function  is such that

 

When you take the second derivative of the function , you obtain 

What can you conclude about the function at ?

Possible Answers:

The point is an absolute maximum.

The point is a local maximum.

The point is a local minimum.

The point is an absolute minimum.

The point is an inflection point.

Correct answer:

The point is an inflection point.

Explanation:

We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.

Plug in 0 into the second derivative to obtain 

So the point is an inflection point.

Example Question #2 : Derivatives

Consider the function 

Find the maximum of the function on the interval .

Possible Answers:

Correct answer:

Explanation:

Notice that on the interval , the term  is always less than or equal to . So the function is largest at the points when . This occurs at  and .

Plugging in either 1 or 0 into the original function  yields the correct answer of 0.

Example Question #3 : Derivatives

In what -intervals are the relative minimum and relative maximum for the function below?

Possible Answers:

Correct answer:

Explanation:

A cubic function will have at most one relative minimum and one relative maximum.  We can determine the zeros be factoring at .  From then we only need to determine if the graph is positive or negative in-between the zeros. 

The graph is positive between  and  (plug in ) and negative between 0 and 4 (plug in ).  This can also be seen from the graph.

 

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Example Question #4 : Derivatives

What is the minimum of the function ?

Possible Answers:

Correct answer:

Explanation:

The vertex form of a parabola is:

where  is the vertex of the parabola.

The function for this problem can be simplified into vetex form of a parabola: 

 ,

with a vertex at .

Since the parabola is concave up, the minimum will be at the vertex of the parabola, which is at .

Example Question #2 : Graphing Functions

What is the y-intercept of the following equation?

Possible Answers:

Correct answer:

Explanation:

The y-intercept can by found by solving the equation when x=0. Thus,

Example Question #3 : Graphing Functions

Determine the y intercept of , where  .

Possible Answers:

Correct answer:

Explanation:

In order to determine the y-intercept of , set 

Solving for y, when x is equal to zero provides you with the y coordinate for the intercept. Thus the y-intercept is .

Example Question #4 : Graphing Functions

What is the -intercept of the function, 

?

Possible Answers:

Correct answer:

Explanation:

To find the -intercept we need to find the cooresponding  value when

Substituting  into our function we get the following:

Therefore, our -intercept is .

Example Question #1 : Graphing Quadratic Functions

What is the value of the -intercept of ?

Possible Answers:

The graph does not have a -intercept

Correct answer:

Explanation:

To find the -intercept we need to find the cooresponding  value when . Therefore, we substitute in  and solve:

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