How do you graph a hyperbola?

To graph a hyperbola, identify the center, vertices, and asymptotes. Plot the center, draw the vertices based on 'a' and 'b', and sketch the asymptotes using the slopes derived from the equation.

What is the difference between a hyperbola and an ellipse?

A hyperbola consists of two separate curves (branches) and has a standard form of $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$ or $$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$$ , while an ellipse is a closed curve with a standard form of $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ .

What is the formula for the area of an ellipse?

The area of an ellipse is given by the formula: $$\text{Area} = \text{π} \times a \times b$$ , where 'a' and 'b' are the semi-major and semi-minor axes, respectively.

What does the term 'asymptote' refer to in hyperbolas?

Asymptotes are lines that the branches of a hyperbola approach but never touch. They provide a guide for the shape of the hyperbola.

What is the property of logarithms that allows for the multiplication of arguments?

The property states that $$\text{log}_b(m \times n) = \text{log}_b(m) + \text{log}_b(n)$$ , allowing the multiplication of arguments to be expressed as the sum of their logarithms.

How do you determine the direction of a hyperbola?

The direction of a hyperbola is determined by the placement of the terms in its equation: 1. If the x-term is positive, the hyperbola opens left and right. 2. If the y-term is positive, the hyperbola opens up and down.

What is the significance of the number '4' in the expression $$4\text{log}_5k - \text{log}_5d$$ ?

The number '4' indicates that the logarithm of 'k' is multiplied by 4, which can be expressed as $$\text{log}_5(k^4)$$ , allowing for the application of logarithmic properties to condense the expression.

What is the standard form of the equation of an ellipse?

The standard form of the equation of an ellipse is: 1. Horizontal ellipse: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ 2. Vertical ellipse: $$\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$$ where (h,k) is the center.

Test 6 Flashcards

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1.

FLASHCARD QUESTION

Front

What is the standard form of the equation of a hyperbola?

Back

2.

FLASHCARD QUESTION

Front

What does the term 'center' refer to in the context of conic sections?

Back

3.

FLASHCARD QUESTION

Front

How do you identify the lengths of a and b in the equation of an ellipse?

Back

4.

FLASHCARD QUESTION

Front

What is the process to expand logarithmic expressions?

Back

5.

FLASHCARD QUESTION

Front

What is the formula for condensing logarithmic expressions?

Back

6.

FLASHCARD QUESTION

Front

What is the significance of the coefficients in the equation of a hyperbola?

Back

7.

FLASHCARD QUESTION

Front

What is the relationship between logarithms and exponents?

Back

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