Math

How do you find the domain and range of a quadratic function?To find the domain of a quadratic function, identify the set of all possible input values (x-values). For any quadratic function, the domain is all real numbers (-∞, ∞). To find the range, determine the vertex of the parabola, which is either the maximum or minimum point. If the parabola opens upwards (a > 0), the range is [k, ∞). If it opens downwards (a < 0), the range is (-∞, k], where k is the y-coordinate of the vertex.

How do you interpret the results of a multiple regression analysis and assess the significance of each predictor variable?To interpret multiple regression results, examine the coefficients to understand the relationship between predictors and the outcome. Assess significance using p-values; predictors with p-values less than 0.05 are typically considered significant. Also, review R-squared for model fit and check for multicollinearity among predictors.

How do you solve a linear equation with variables on both sides and unknowns include negative fractions and integers?To solve a linear equation with variables on both sides and unknowns including negative fractions and integers, first simplify both sides by combining like terms. Move all variable terms to one side and constant terms to the other. Then, isolate the variable by performing inverse operations, carefully handling negative fractions and integers.

How do you solve the equation 3x^2 – 2x + 1 = 0 using the quadratic formula?To solve the equation 3x^2 – 2x + 1 = 0 using the quadratic formula, use x = (-b ± √(b² – 4ac)) / 2a. Here, a = 3, b = -2, and c = 1. Calculate the discriminant (b² – 4ac): (-2)² – 4(3)(1) = 4 – 12 = -8. Since the discriminant is negative, the equation has two complex solutions: x = (2 ± √(-8)) / 6 = (2 ± 2i√2) / 6 = 1/3 ± i√2/3.

How do you find the volume of a solid of revolution using the disk method?To find the volume of a solid of revolution using the disk method, integrate the area of circular disks perpendicular to the axis of rotation. For a function y=f(x) rotated around the x-axis from x=a to x=b, the volume V is given by V = π∫[a to b] (f(x))^2 dx.

How do you calculate and interpret the correlation coefficient in a data set?The correlation coefficient, denoted as ‘r’, quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. Calculate it using the formula: r = Σ[(Xi – X̄)(Yi – Ȳ)] / [√Σ(Xi – X̄)² * Σ(Yi – Ȳ)²]. An ‘r’ close to 1 or -1 indicates a strong relationship, while an ‘r’ near 0 indicates a weak or no linear relationship.

How do you determine if two triangles are similar with given angles and side lengths?To determine if two triangles are similar, you can use the Angle-Angle (AA) criterion, where two corresponding angles are equal, the Side-Angle-Side (SAS) criterion, where two sides are proportional and the included angle is equal, or the Side-Side-Side (SSS) criterion, where all corresponding sides are proportional.

In a system where you have a sphere inscribed within a right circular cylinder and another sphere that circumscribes this cylinder, if the sphere inscribed in the cylinder has a radius r, find the ratio of the volume of the larger sphere to the volume of The sphere inscribed in the cylinder has a radius r and thus a volume of (4/3)πr^3. The larger sphere circumscribing the cylinder has a diameter equal to the cylinder’s diagonal, which is 2√2r, giving it a radius of √2r and a volume of (4/3)π(√2r)^3. The ratio of the volumes is (√2)^3 = 2√2.

If four-fifths of a number is increased by ten, the result is five more than three-fourths of the original number. What is the original number?Let the original number be x. According to the problem, (4/5)x + 10 = (3/4)x + 5. Solving this equation, we find that x = 100. Therefore, the original number is 100.

How can we differentiate between Type I and Type II errors, and which degree of error is more acceptable in medical testing outcomes for rare diseases?Type I error, or false positive, occurs when a test incorrectly indicates the presence of a condition. Type II error, or false negative, occurs when a test fails to detect a condition. In medical testing for rare diseases, minimizing Type II errors is generally more acceptable, as missing a diagnosis can have more severe consequences than a false positive.