Which of the following equations is of a parabola with a vertex at (1, 2)?
A parabola can be described by the standard form of its equation, which depends on its vertex. The vertex form of a parabola is given by the equation: y = a(x – h)² + k In this equation, (h, k) represents the vertex of the parabola. For a parabola with a vertex at (1, 2), […]
How to Find the Exact Location of All the Relative and Absolute Extrema of the Function f(x) = x³ – 6x² + 9x – 2?
To find the extrema of the function f(x) = x³ – 6x² + 9x – 2, we first need to determine the critical points by calculating its derivative and setting it to zero. The first derivative of the function is: f'(x) = 3x² – 12x + 9 Next, we set the derivative equal to zero: […]
Find the Correct Sum of the Polynomials 4x³ + 2x + 9 and 2x³ + 5x + 3
To find the sum of the given polynomials, we need to combine like terms. Starting with the first polynomial, we have: 4x³ 2x 9 Now, for the second polynomial: 2x³ 5x 3 Now we add these polynomials together: 4x³ + 2x + 9 + 2x³ + 5x + 3 Next, let’s group the like terms: […]
Find the vertex and axis of symmetry and intercepts for a quadratic equation y = x² + 6x + 5
To analyze the quadratic equation y = x² + 6x + 5, we need to find the vertex, axis of symmetry, and intercepts. 1. Vertex The vertex of a quadratic function in the standard form y = ax² + bx + c can be found using the formula: x = -b / (2a) In our […]
The product of two consecutive positive integers is 55 more than their sum. Find the integers.
Let the two consecutive positive integers be x and x + 1. According to the problem, the product of these integers can be expressed as: x(x + 1) Their sum can be expressed as: x + (x + 1) = 2x + 1 The problem states that the product is 55 more than their sum, […]
a and b are vertical angles with ma x and mb 5x 80 what is ma
To find the value of ma, we first need to understand that vertical angles are equal. This means that the angles ma and mb have the same measure. We are given that mb = 5x – 80. According to the properties of vertical angles, we can set ma = mb: ma = 5x – 80 […]
Find the volume of the solid that lies within the cylinder x² + y² ≤ 1 and the sphere x² + y² + z² ≤ 4
To find the volume of the solid that lies within both the cylinder defined by the equation x² + y² ≤ 1 and the sphere defined by x² + y² + z² ≤ 4, we need to set up the problem in cylindrical coordinates. In cylindrical coordinates, we make the substitutions: x = r cos(θ) […]
A die with 6 sides is rolled. What is the probability of rolling a number less than 5?
When rolling a standard six-sided die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. To find the probability of rolling a number less than 5, we first identify which numbers on the die meet this criterion. The numbers less than 5 are 1, 2, 3, and 4. Counting these, we […]
If f(1) = 0, what are all the roots of the function f(x) = x^3 + 3x^2 + x + 3? Use the Remainder Theorem.
To find the roots of the function f(x) = x^3 + 3x^2 + x + 3 using the Remainder Theorem, we first verify that f(1) = 0, which indicates that x = 1 is one of the roots of the function. Let’s evaluate f(1): f(1) = 1^3 + 3(1^2) + 1 + 3 = 1 […]
If you roll two fair six-sided dice, what is the probability that the sum is 5 or lower?
To find the probability of rolling a sum of 5 or lower with two six-sided dice, we first need to determine the total possible outcomes when rolling the two dice. Each die has 6 sides, so when rolling two dice, the total combinations are: Total Outcomes = 6 (sides on die 1) × 6 (sides […]