🧮 algebra

1. **Problem 1: Identify invariant points under $g(x) = -2f(x)$ given $y = f(x)$ and invariant point $(-1,0)$.** - An invariant point $(a,b)$ satisfies $g(a) = b$.

1. **State the problem:** We are given a piecewise function defined as: $$f(x) = \begin{cases} x^2 & \text{if } x \geq 1 \\ -x^2 + 1 & \text{if } x < 0 \end{cases}$$

1. **State the problem:** Simplify the expression $$(\sqrt{10} - \sqrt{12})(12\sqrt{10} - 10\sqrt{12})$$. 2. **Use the distributive property (FOIL) to expand:**

1. The problem is to simplify the fraction $\frac{73527271}{2}$. 2. The formula for division is $\text{Dividend} \div \text{Divisor} = \text{Quotient}$.

1. **Problem Statement:** We start with the basic square root function:

1. **State the problem:** We start with the function $f(x) = x^2$ and apply two transformations: shift upward by 52 units and shift to the right by 97 units. 2. **Recall the transf

1. **State the problem:** We want to rewrite the quadratic equation $$y = x^2 + 12x + 27$$ in vertex form $$y = (x - h)^2 + k$$ where $(h, k)$ is the vertex of the parabola. 2. **R

1. **Problem 1:** Given $f(x) = -3x^2 + 9x + 54$, find the x-intercepts, y-intercept, vertex, and line of symmetry. 2. **Find x-intercepts:** Set $f(x) = 0$:

1. **State the problem:** We need to determine which statement about vertical lines is true. 2. **Recall the definition of slope:** The slope $m$ of a line is given by the formula

1. **State the problem:** We need to find the equation of a line that passes through the point $(-7, 5)$ and has a slope of zero. 2. **Recall the rule for slope zero:** A line with

1. **State the problem:** We need to find which expression among A, B, C, and D is equivalent to $42+70$. 2. **Calculate the sum:**

1. **State the problem:** We need to expand the cubic polynomial $f(x) = x^3 - [blank]x^2 + [blank]x - [blank]$ by filling in the blanks with appropriate coefficients. 2. **General

1. **State the problem:** We need to find the least common multiple (LCM) of the numbers $4$ and $5$. 2. **Recall the formula for LCM:** The LCM of two numbers $a$ and $b$ can be f

1. The problem is to expand the cubic polynomial function $$f(x) = x^3 - \text{[blank]}x^2 + \text{[blank]}x - \text{[blank]}$$ where the coefficients are missing. 2. To expand or

1. The problem asks to find the Least Common Multiple (LCM) of given pairs of numbers: $4$ and $5$, $6$ and $8$, $4$ and $10$, $5$ and $8$, $8$ and $20$. 2. The LCM of two numbers

1. The problem is to find the Least Common Multiple (LCM) of $8$ and $10$. 2. The LCM of two numbers is the smallest positive integer that is divisible by both numbers.

1. **State the problem:** We want to find the number of days until the museum performs both a school show (which happens every Monday) and a library show (which happens every 5 day

1. **State the problem:** We need to find an equation that represents the locations of the outer edges of a stage centered at the 35-yard line, with the distance from the center to

1. **State the problem:** We want to find the complex numbers $z = a + bi$ such that $$z^2 = 3 - 4i$$ where $a$ and $b$ are real numbers.

1. **State the problem:** We need to perform the division $$\frac{-5}{2} \div -3$$ and simplify the result. 2. **Recall the rule for dividing fractions:** Dividing by a number is t

1. **State the problem:** Find the equation of a line in slope-intercept form that passes through the point $(-2, 4)$ and is parallel to the line $y = 9$. 2. **Recall the slope-int