Algebra II : Quadratic Functions

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #6 : Hyperbolic Functions

What is the value of ?

Possible Answers:

Correct answer:

Explanation:

The hyperbolic tangent will need to be rewritten in terms of hyperbolic sine and cosine.

According to the properties:

Therefore:

Example Question #7 : Hyperbolic Functions

Simplify:  

Possible Answers:

Correct answer:

Explanation:

The following is a property of hyperbolics that is closely similar to the problem.

We will need to rewrite this equation by taking a negative one as the common factor, and divide the negative one on both sides. 

Substitute the value into the problem.

 

 

Example Question #8 : Hyperbolic Functions

Which of the following is the correct expression for a hyperbola that is shifted   units up and   to the right of ?

Possible Answers:

Correct answer:

Explanation:

The parent function of a hyperbola is represented by the function  where  is the center of the hyperbola. To shift the original function up by   simply add . To shift it to the right   take away .

Example Question #9 : Hyperbolic Functions

Find the foci of the hyperbola:  

Possible Answers:

Correct answer:

Explanation:

Write the standard forms for a hyperbola.

OR:

The standard form is given in the second case, which will have different parameters compared to the first form.

Center:  

Foci:  , where 

Identify the coefficients  and substitute to find the value of .

The answer is:  

Example Question #10 : Hyperbolic Functions

Given the hyperbola , what is the  value of the center?

Possible Answers:

Correct answer:

Explanation:

In order to determine the center, we will first need to rewrite this equation in standard form.  

Isolate 41 on the right side.  Subtract  and add  on both sides.

The equation becomes:

Group the x and y terms.  Be careful of the negative signs.

Pull out a common factor of 4 on the second parentheses.

Complete the square twice.  Divide the second term of each parentheses by two and square the quantity.  Add the terms on both sides.

This equation becomes:

Factorize the left side and simplify the right.

Divide both sides by nine.

The equation is now in the standard form of a hyperbola.

The center is at:  

The answer is:  

Example Question #11 : Hyperbolic Functions

Which of the following shapes does the graph of the equation  take?

Possible Answers:

Circle

Parabola

Ellipse

Hyperbola

Correct answer:

Ellipse

Explanation:

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . Both include x- and y-terms combined using subtraction. The equation for an ellipse is . Because the given equation connects the x- and y-terms using addition rather than subtraction, it represents an ellipse rather than a hyperbola. If the equation took the form , it would represent a circle. If the equation took the form , it would represent a parabola.

 

 

Example Question #101 : Quadratic Functions

Which of the following shapes does the graph of the equation  take? 

Possible Answers:

Hyperbola

Ellipse

Parabola

Circle

Correct answer:

Hyperbola

Explanation:

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . Both include x- and y-terms combined using subtraction. If the equation took the form  (using addition rather than subtraction to combine the x- and y-terms), it would represent an ellipse. If the equation took the form , it would represent a circle. If the equation took the form , it would represent a parabola.

 

 

Example Question #13 : Hyperbolic Functions

What are the coordinates of the center of the hyperbolic inequality ?

Possible Answers:

Correct answer:

Explanation:

 The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both, (h, v) is the center of the hyperbola. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. Because the two terms are combined using subtraction and the y-term appears first, this inequality represents a vertical hyperbola. To derive the center of a hyperbola from its equation or inequality, flip the sign of the constants that appear after the x and y in the equation or inequality. The constant following x is -1, so the x-coordinate of the center is 1. The constant following y is 2, so the y-coordinate of the center is -2. 

 

The graph of the hyperbolic inequality  appears as follows:

Hyperbola2

With vertices of (1, 1) and (1, -5), you can see that the midpoint between them is (1, -2).

Example Question #12 : Hyperbolic Functions

What are the coordinates of the center of the hyperbolic inequality ?

Possible Answers:

Correct answer:

Explanation:

The equation for a horizontal hyperbola is   . The equation for a vertical hyperbola is . In both, (h, v) is the center of the hyperbola. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. Because the two terms are combined using subtraction and the x-term appears first, this inequality represents a horizontal hyperbola. To derive the center of a hyperbola from its equation or inequality, flip the sign of the constants that appear after the x and y in the equation or inequality. The constant following x is 3, so the x-coordinate of the center is -3. No constant follows y, so the y-coordinate of the center is 0. 

 

The graph of the hyperbolic inequality   appears as follows: 

 Hyperbola1

As you can see, the midpoint between the two vertices of (-6, 0) and (0, 0) is (-3, 0).

Example Question #502 : Functions And Graphs

Which equation does this graph represent?

Screen shot 2020 08 26 at 8.10.13 am

Possible Answers:

Correct answer:

Explanation:

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both, (h, v) is the center of the hyperbola. The graph shows a horizontal hyperbola, so in its corresponding equation the x-term must appear first. The center lies at (-2, -1), so x must be followed by the constant 2, and y must be followed by the constant 1. The graph shows a hyperbola rather than an ellipse, so the x- and y-terms must be combined using subtraction rather than addition.

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