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To find the coordinates of the vertex for the quadratic function given by the equation f(x) = 2x² + 4x + 9, we can use the vertex formula. The vertex (h,k) of a quadratic function of the standard form f(x) = ax² + bx + c can be found using the following formulas: h = … Read more

To solve the system of equations given by 7x + 2y = 4 and 5y + 3x = 10, we can use the method of substitution or elimination. In this solution, we will employ the elimination method for clarity. First, let’s rewrite the equations clearly: Equation 1: 7x + 2y = 4 Equation 2: 3x … Read more

To find the value of f(g(3)), we first need to calculate g(3). The function g(x) is defined as: g(x) = x2 – 10 Now, substitute 3 into the function: g(3) = 32 – 10 g(3) = 9 – 10 g(3) = -1 Now that we have g(3) = -1, we will substitute this value into … Read more

The equation of a circle centered at the origin (0, 0) with a radius of 5 units is given by: x² + y² = r² Here, r represents the radius of the circle. In this case, the radius is 5, so we substitute it into the equation: x² + y² = 5² This simplifies to: … Read more

Finding the Inverse Function To find the inverse function of f(x) = 5x + 4, we need to follow these steps: Start by replacing f(x) with y: y = 5x + 4 Next, swap x and y: x = 5y + 4 Now, solve for y: Subtract 4 from both sides: x – 4 = … Read more

When we talk about symmetry with respect to the origin, we’re referring to a characteristic of certain graphs. Specifically, a graph is symmetric about the origin if it satisfies the condition that if a point (x, y) lies on the graph, then the point (-x, -y) also lies on the graph. This property is often … Read more

To find the value of m7, we first need to understand the relationship between the angles formed by parallel lines. When two lines are parallel, certain pairs of angles are either congruent or supplementary based on their positions with respect to a transversal. In this case, let’s assume that m4 represents an angle formed by … Read more

To find P(A | B), the conditional probability of event A given event B, we utilize the definition of conditional probability which states: P(A | B) = P(A and B) / P(B) For independent events, we know that: P(A and B) = P(A) * P(B) Given that: P(A) = 0.38 P(B) = 0.55 We can … Read more

To find the arc length function for the curve defined by the equation y = 2x^(3/2), we will follow these steps: Understand the Arc Length Formula: The arc length L of a curve y = f(x) from x = a to x = b is given by: L = ∫ from a to b √(1 … Read more

To find the dimensions of a rectangle that maximizes its area given a fixed perimeter of 100 meters, we can use the properties of rectangles and some basic algebra. First, let's denote the length of the rectangle as L and the width as W. The formula for the perimeter (P) of a rectangle is: P … Read more