Modeling #8 - Questions and Answers

Question 1 To explain: a. How to determine the length of the transverse axis by standard equation of a hyperbola. b. How to determine whether the transverse axis is vertical or horizontal Answer Given: The standard equation of a hyperbola. Calculation: a. Let us consider the standard equation of the hyperbola passing through the origin (0,0) to be of the form, x²/a² - y²/b² = 1, with asymptotes y = ±(b/a)x, vertices (±a, 0), and foci (±√(a² + b²), 0) The transverse axis is the line segment connecting the vertices of the hyperbola. Here, as the vertices are (±a, 0), therefore the length of the transverse axis is 2a. b. If the standard equation of the hyperbola passing through the origin (0,0) to be of the form, x²/a² - y²/b² = 1, with asymptotes y = ±(b/a)x, vertices (±a, 0), and foci (±√(a² + b²), 0) Then the hyperbola is said to have a horizontal transverse axis. And, If the standard equation of the hyperbola passing through the origin (0,0) to be of the form, y²/a² - x²/b² = 1, with asymptotes y = ±(a/b)x, and foci (0, ±√(a² + b²)) Then the hyperbola is said to have a vertical transverse axis. Conclusion: a. Length of the transverse axis is 2a. b. If the hyperbola is in the form x²/a² - y²/b² = 1 then the hyperbola is said to have a horizontal transverse axis and if the hyperbola is in the form y²/a² - x²/b² = 1 then the hyperbola is said to have a vertical transverse axis. Question 2 The interval notation for 4 ≤ x < 19. Answer Calculation: The given inequality implies that x is greater than or equal to 4 on the left side, so we use a closed bracket at 4. And on the right side, it is less than 19, so we use an open parenthesis at 19, because 19 is not included. Hence the interval notation is: [4, 19). Conclusion: The interval notation is given by: [4, 19). Question 3 a) The interval notation for the expression x ≤ 5. Answer Interval notation is a simplified form of writing the solution to an inequality or system of inequalities, using the bracket and parenthesis symbols in lieu of the inequality symbols. Intervals with parentheses are called open intervals, meaning the variable cannot have the value of the endpoints. Formula used: Inequality Interval notation Description x > a (a, ∞) x is greater than a x < a (-∞, a) x is less than a x ≥ a [a, ∞) x is greater than or equal to a x ≤ a (-∞, a] x is less than or equal to a a < x < b (a, b) x is strictly between a and b a ≤ x < b [a, b) x is between a and b, including a a < x ≤ b (a, b] x is between a and b, including b a ≤ x ≤ b [a, b] x is between a and b, including a and b Conclusion: The interval notation for the expression x ≤ 5 is given by (-∞, 5]. Question 4 The difference quotient for the f(x) = mx + b function Answer Formula used: The difference quotient of the function f(x) is given by: Difference quotient = (f(x+h) - f(x)) / h Calculation: First of all, we find an expression for f(x+h). In order to find the expression for f(x + h), we replace x by x + h in f(x). f(x+h) = m(x+h) + b f(x+h) = mx + mh + b We obtain the following when we enter the expressions for f(x+h) and f(x) into the difference quotient formula: Difference quotient = (mx + mh + b - (mx + b)) / h Difference quotient = (mx + mh + b - mx - b) / h Difference quotient = (mh) / h Difference quotient = m. Conclusion: On plugging the given linear function in the difference quotient formula, we get the difference quotient of the function f(x) = mx + b as m.