🧮 algebra

1. **State the problem:** We need to express $g(x)$ in terms of $f(x)$ given that the graph of $g(x)$ is the same as $f(x)$ but shifted upward by 8 units. 2. **Recall the vertical

1. **State the problem:** We are given two sinusoidal functions $g(x)$ and $f(x)$ where $g(x)$ is the blue graph shifted 8 units up from $f(x)$, which is the red graph. We want to

1. **State the problem:** We need to find the ratio of the rise to the run of the line segment TX, where T has coordinates $(15, 2)$ and X has coordinates $(45, 5)$. 2. **Recall th

1. **State the problem:** We need to find whether the slope of the line passing through points $T(15, 2)$ and $X(45, 5)$ is positive or negative. 2. **Formula for slope:** The slop

1. **State the problem:** We need to determine which graph correctly represents the polynomial $$f(x) = 2x^3(x - 2)^2$$. 2. **Analyze the polynomial:**

1. **Problem statement:** We are given the function $$y = 5 - |x - 2|$$ and need to:

1. **State the problem:** We start with a solution that is $80\%$ milk and $20\%$ water. After adding some water, the milk to water ratio becomes $8:5$. Then, $39$ L of this new so

1. The problem is to evaluate the expression $2 + \sqrt{25}$.\n\n2. Recall that the square root function $\sqrt{x}$ gives the non-negative number which, when squared, equals $x$.\n

1. **State the problem:** Solve the equation $1 + 2(4x - 9) = x$ for $x$. 2. **Use the distributive property:** Multiply $2$ by each term inside the parentheses:

1. **State the problem:** Solve for $x$ in the equation $2x + 4x = 9$. 2. **Combine like terms:** Since both terms on the left side contain $x$, add their coefficients:

1. **State the problem:** Solve the quadratic equation $$x^2 + 3 = 0$$. 2. **Formula and rules:** To solve for $x$, isolate $x^2$ and then take the square root of both sides. Remem

1. Let's start by stating the problem: We want to find an equation that represents the sequence 2.5, 2.73, 2.95. 2. The sequence appears to be increasing, so we can check if it fit

1. **State the problem:** We need to find a route visiting Bank, Post Office, and Dry Cleaners starting and ending at Home, with total distance less than $12.5$ miles. 2. **Given d

1. **State the problem:** We need to find the order in which the runners finished the race based on the given time differences. 2. **Define variables:** Let $B$ be the finishing ti

1. **State the problem:** We need to find the profit on six dozen calculators. 2. **Identify given information:**

1. **State the problem:** We want to find how much money is saved by paying $900 upfront instead of paying $50 down plus $100 per month for 14 months. 2. **Identify the formula:**

1. **State the problem:** A parking garage charges $2.50$ for the first hour and $0.50$ for each additional hour. A customer parked from 10 A.M. to 3 P.M. and gave $45$ to the atte

1. **State the problem:** We need to find the extra amount received by the salesperson for selling 216 items, given that they receive $10 for every item sold beyond 200 items.

1. **State the problem:** We are given the quadratic equation $$y = 3x^2 + 6x + 3$$ and a graph of this parabola. We need to determine which has the smaller minimum value: the equa

1. **State the problem:** We are given the function $$f(x) = 3(x - 1)^2 + 1$$ and a graph of a parabola with vertex at $$(-4, 0)$$. We need to compare the minimum values of the equ

1. **Problem Statement:** Find the value of $x$ in the equation $$2x + 3 = 11$$.