Determine if the graph is symmetric about the x axis, the y axis, or the origin: r = 4 cos(3θ)
To determine the symmetry of the polar graph given by the equation r = 4 cos(3θ), we need to check the conditions for symmetry about the x-axis, y-axis, and the origin. 1. Symmetry about the x-axis: A polar graph is symmetric about the x-axis if replacing θ with -θ yields the same r value. Let’s […]
What is the equation of the line that passes through (4, 1) and (2, 3)?
To find the equation of the line that passes through the points (4, 1) and (2, 3), we can follow these steps: Calculate the slope (m): The slope formula is given by: m = (y2 – y1) / (x2 – x1) Plugging in our points (x1, y1) = (4, 1) and (x2, y2) = (2, […]
How to Find a Vector Equation and Parametric Equations for a Line Segment Connecting Two Points?
To find the vector equation and parametric equations for the line segment that joins the points P(0, 1, 1) and Q(12, 13, 14), we can follow these steps: Step 1: Identify the Direction Vector First, we need to determine the direction vector, which is obtained by subtracting the coordinates of point P from those of […]
Find the area of a sector with an arc length of 60 in and a radius of 15 in
To find the area of a sector given the arc length and radius, we first need to determine the angle in radians of the sector. The formula for arc length (s) in terms of the radius (r) and angle (θ in radians) is: s = r * θ Given that the arc length (s) is […]
Consider the Parametric Equations Below: x = t^2, y = t^9. How Can I Sketch the Curve Using These Equations to Plot Points?
To sketch the curve defined by the parametric equations x = t2 and y = t9, we can start by plotting several points by substituting different values of t. Let’s choose a range of t values, both positive and negative, to see how the curve behaves: For t = -2: x = (-2)2 = 4, […]
If f(x) = x² + 25 and g(x) = x + 5, what is the domain of f(g(x)) and find f(g(1))?
To determine the domain of the composite function f(g(x)), we first need to look at the domains of both functions f(x) and g(x). The function f(x) = x² + 25 is a polynomial function and is defined for all real numbers, meaning its domain is (−∞, ∞). Next, let’s consider the function g(x) = x […]
What is the quotient when x³ + 3x² + 3x + 2 is divided by x² + x + 1?
To find the quotient when dividing the polynomial x³ + 3x² + 3x + 2 by x² + x + 1, we can use polynomial long division. The first step is to divide the leading term of the numerator, x³, by the leading term of the denominator, x², which gives us x. We then multiply […]
What is 3 more than the product of 8 and a number?
To find out what is 3 more than the product of 8 and a number, let’s break the problem down step by step. First, let’s define our variable. Let’s call the number we are referring to ‘x’. The phrase ‘the product of 8 and a number’ means we multiply 8 by our variable ‘x’. Mathematically, […]
Give a Recursive Definition of the Sequence a_n = n * a_{n-1} for n ≥ 1
To define the sequence a_n recursively, we need to establish a base case and a recursive case. The given sequence is defined as follows: Base Case: When n = 1, we set a_1 = 1. Recursive Case: For n > 1, we define a_n in terms of a_{n-1} with the formula: a_n = n * […]
How to Eliminate the Parameter in the Equation x = 5t, y = t + 8?
To eliminate the parameter from the equations given, we start with the two equations: x = 5t y = t + 8 First, we solve the first equation for the parameter t: t = x / 5 Next, we substitute this expression for t into the second equation: y = (x / 5) + 8 […]