Which of the following describes the translation of y = x to y = x + 7?
The translation of the equation y = x to y = x + 7 represents a vertical shift of the graph of the equation. In this case, we are adding 7 to the y-value for every x-value. This means that for each point (x, y) on the original line y = x, the new point […]
What is the property of equality that justifies the statement ‘if a = b, then b = a’?
The property of equality that justifies the statement ‘if a = b, then b = a’ is known as the Symmetric Property of Equality. This property states that if two values are equal, then they can be expressed in either order. In other words, if you have two quantities, say ‘a’ and ‘b’, and you […]
If a polynomial function f(x) has roots 8, 1, and 6i, what must also be a root of f(x)?
Since the polynomial function has a complex root, it is important to note that complex roots occur in conjugate pairs when dealing with polynomials with real coefficients. In this case, the root 6i has a conjugate of -6i. Therefore, if 6i is one of the roots, -6i must also be a root of the polynomial […]
If y equals 32 then what is x in this equation y = 192 x?
To solve for x in the equation y = 192x, we first substitute the value of y with 32. So, we have: 32 = 192x Next, we need to isolate x. To do this, we divide both sides of the equation by 192: x = 32 / 192 Now, we simplify the fraction: x = […]
Can 72 and 20 respectively be the LCM and HCF of two numbers? Give reason.
No, 72 and 20 cannot be the least common multiple (LCM) and highest common factor (HCF) of the same two numbers. The reason for this lies in the relationship between LCM and HCF. For any two numbers, the product of their LCM and HCF is equal to the product of the two numbers themselves. Mathematically, […]
How do you simplify sine squared theta divided by the quantity 1 minus sine squared theta?
To simplify the expression sin²(θ) / (1 – sin²(θ)), we start by recognizing a fundamental trigonometric identity. We know that: 1 – sin²(θ) = cos²(θ) Using this identity, we can substitute cos²(θ) for 1 – sin²(θ) in the original expression: sin²(θ) / (1 – sin²(θ)) = sin²(θ) / cos²(θ) This expression can be rewritten using […]
Explain Circle with the Help of Conic Sections
A circle is a special type of conic section that forms when a plane intersects a cone parallel to the cone’s base. In geometric terms, a circle is defined as the set of all points that are at a fixed distance, called the radius, from a central point, known as the center. To visualize this, […]
If a Polynomial Function f(x) has Roots 4, 13i, and 5, What Must be a Factor of f(x)?
If a polynomial function f(x) has roots at 4, 13i, and 5, then one of its factors is derived from these roots. In polynomial functions, complex roots always occur in conjugate pairs. Since we have 13i as a root, its conjugate -13i must also be a root. Therefore, we can list the roots as 4, […]
How can we simplify the rational expression \( \frac{t^2}{6t^2 + 9} \) and what are the restrictions on the variable?
To simplify the rational expression \( \frac{t^2}{6t^2 + 9} \), we start by looking at the denominator. First, we can factor the denominator. The expression in the denominator is \( 6t^2 + 9 \). Notice that both terms have a common factor of 3. We can factor this out: \[ 6t^2 + 9 = 3(2t^2 […]
What is the sum of the finite arithmetic series 26, 29, 32, 35, 38, 41, 44?
The sum of the finite arithmetic series can be calculated using the formula: S = n/2 * (a + l) S = sum of the series n = number of terms a = first term l = last term In this series, the first term a is 26, and the last term l is 44. […]