Math

How do you find the transformations applied to the graph of a function f(x) represented by f(x) = -2(x – 4)² + 3?To find the transformations applied to the graph of f(x) = -2(x – 4)² + 3, identify the following: horizontal shift 4 units right (x – 4), vertical stretch by a factor of 2, reflection across the x-axis (-2), and vertical shift 3 units up (+3).

How can you use the unit circle to determine the sine and cosine values of special angles?The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Special angles, typically 0°, 30°, 45°, 60°, and 90°, correspond to specific points on this circle. The x-coordinate of these points represents the cosine value, while the y-coordinate represents the sine value. For example, at 0°, the coordinates are (1, 0), so cos(0°) = 1 and sin(0°) = 0. At 45°, the coordinates are (√2/2, √2/2), so cos(45°) = √2/2 and sin(45°) = √2/2. This method can be applied to other special angles to find their sine and cosine values.

What is the largest prime number less than 50?The largest prime number less than 50 is 47. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime numbers less than 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.

If a train travels at a constant speed of 50 miles per hour, how far will it travel in 3 hours?To determine the distance traveled by a train moving at a constant speed of 50 miles per hour over 3 hours, use the formula: Distance = Speed × Time. Therefore, the train will travel 50 miles/hour × 3 hours = 150 miles.

How do you find the derivative of a function using the chain rule?To find the derivative of a composite function using the chain rule, identify the outer function and the inner function. Differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function. Mathematically, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

How do you use the method of integration by parts to solve the integral of the product of two functions?Integration by parts is a technique derived from the product rule for differentiation. For functions u(x) and v(x), it states: ∫u(x)v'(x)dx = u(x)v(x) – ∫v(x)u'(x)dx. Choose u(x) and v'(x) such that the resulting integral ∫v(x)u'(x)dx is simpler to evaluate. Apply the formula iteratively if necessary.

What are the properties and basic identities of trigonometric functions?Trigonometric functions, including sine, cosine, tangent, cotangent, secant, and cosecant, have fundamental properties and identities. Key properties include periodicity, symmetry, and boundedness. Basic identities include Pythagorean identities (e.g., sin²θ + cos²θ = 1), angle sum and difference identities, double-angle identities, and reciprocal identities.

How can I prove trigonometric identities involving double-angle and half-angle formulas?To prove trigonometric identities involving double-angle and half-angle formulas, use fundamental identities such as sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) – sin²(θ). For half-angle formulas, use sin(θ/2) = ±√((1 – cos(θ))/2) and cos(θ/2) = ±√((1 + cos(θ))/2). Simplify expressions and verify both sides of the identity.

How do you derive and apply the Double Angle Formula for sine and cosine in trigonometric integration problems?The Double Angle Formulas for sine and cosine are derived from the sum formulas: sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) – sin²(θ). These formulas simplify integrals involving trigonometric functions by allowing substitution, facilitating easier integration.

What is the distributive property of multiplication over addition?The distributive property of multiplication over addition states that multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the products. Mathematically, it is expressed as a(b + c) = ab + ac.