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Solving the System of Equations To solve the system of equations: Equation 1: y = x² + 3 Equation 2: y = x + 5 we can use a substitution method since both equations equal y. Here are the detailed steps: Step 1: Set the equations equal to each other Since both equations are equal … Read more

To determine whether a function is even or odd algebraically, you can follow these straightforward steps: 1. Understanding the Definitions First, it’s essential to grasp what it means for a function to be even or odd: Even functions: A function f(x) is classified as even if it fulfills the condition f(-x) = f(x) for all … Read more

Understanding Amplitude, Period, and Midline The function in question is y = 7 sin(4x) + 2. Let’s break it down to identify its amplitude, period, and midline. Amplitude The amplitude of a sine function is determined by the coefficient in front of the sine function. In this case, the function has a coefficient of 7 … Read more

To determine the greatest common factor (GCF) of 8 and 36, we can use a few different methods, but one of the simplest is to list out the factors of each number and see what they have in common. Step 1: List the factors First, let’s find the factors of each number: Factors of 8: … Read more

To find a quadratic polynomial given its zeros, you can use the fact that if a polynomial has zeros at r and s, it can be expressed in the factorized form: f(x) = (x – r)(x – s) In this case, the zeros are 4 and 2. Plugging these values into the formula gives: f(x) … Read more

To determine how many ways you can arrange 7 books on a shelf, picking 5 at a time, we can use the concept of permutations. This is useful because the order of the books matters when arranging them. The formula for permutations is given by: P(n, r) = n! / (n – r)! Where: n … Read more

Drawing the Rose Curve: r = 4 cos(2θ) The rose curve defined by the equation r = 4 cos(2θ) is a fascinating polar graph that showcases a petal structure. Here’s how to draw this curve step-by-step: Step 1: Understanding the Parameters r = the distance from the origin to the curve. θ = the angle … Read more

To find the quotient when dividing the polynomial x³ + 5x² + 2x + 5 by x – 2, we will use polynomial long division. 1. **Setup:** Write the polynomial you want to divide (the dividend) and the divisor. Here, we’re dividing x³ + 5x² + 2x + 5 by x – 2. 2. **First … Read more

To solve the problem of finding two positive numbers where the sum of the first number and twice the second number equals 56, while maximizing their product, we can approach it using algebra and calculus. Let us define two positive numbers: x = the first number y = the second number From the problem statement, … Read more

The axis of symmetry for a quadratic function in the standard form f(x) = ax² + bx + c can be calculated using the formula: x = -b / (2a) In your case, the quadratic function is: f(x) = 3x² – 18x + 7 Here, the coefficients are: a = 3 b = -18 c … Read more