🧮 algebra

1. **Problema:** Dada la función $$f(x) = -x^3 - 2x^2 + 6$$, hallar los vértices, puntos de corte con los ejes, dominio y rango, y analizar la gráfica. 2. **Tipo de función:** Es u

1. **State the problem:** We are given the height function of a ball as $x(t) = 160t^2 - 40t$, where $x(t)$ is the height in feet and $t$ is the time in seconds after the ball was

1. **State the problem:** Simplify the expression $25 - (x + 3 - x^2)$. 2. **Apply the distributive property:** When subtracting a group of terms inside parentheses, distribute the

1. **State the problem:** Find the x-coordinates where the graphs of \( f(x) = \cos^2(x) + 6\cos(x) \) and \( g(x) = -5 \) intersect for \( 0 \leq x < 2\pi \). 2. **Set the functio

1. The problem asks which function has a minimum at the point $(-2, -3)$. 2. The vertex form of a quadratic function is given by $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex

1. **State the problem:** Find the number of pairs of positive integers $(a,b)$ such that $$a^3 + b^2 = 100.$$\n\n2. **Understand the constraints:** Both $a$ and $b$ are positive i

1. **State the problem:** Solve the equation $$1 + \frac{3}{x - 1} = \frac{2x + 7}{x^2 + x - 2}$$ for $x$. 2. **Identify the denominator factorization:** Note that $$x^2 + x - 2 =

1. **State the problem:** We are given four polynomials:

1. **State the problem:** We are given a table of values for a function $f$ at selected values of $x$ and asked to determine which claim about the function type (exponential or log

1. **State the problem:** We are given several equations involving variables $x$, $b$, and $a$ and need to verify or solve these equations. 2. **Given equations:**

1. The problem asks to find $f^{-1}(4)$, which means we want to find the value of $x$ such that $f(x) = 4$. 2. From the table, we look for the output value 4 in the row of $f(x)$ v

1. **State the problem:** We have a piecewise linear function $f$ defined by two line segments connecting points $(-4,3)$, $(-1,0)$, and $(2,-3)$. We want to find the graph of its

1. **State the problem:** We need to plot the parabola $Q$ given by the equation $$y = \frac{(x+6)(x+2)}{4}$$ and understand its shape and key points. 2. **Formula and explanation:

1. **State the problem:** We are given a cubic function $h(x) = x^3 - 3x$ representing a water slide profile, and two points $A(2,3)$ and $B(6,9)$ defining a line segment. We want

1. **Stating the problem:** We are given quantities of bags and their associated costs in UGX (Ugandan shillings). The problem is to analyze or calculate the cost per bag or total

1. **State the problem:** We need to transport 1200 bags using two types of vehicles: Pickup trucks and Isuzu Diana trucks. 2. **Define variables:** Let $x$ be the number of trips

1. **Restate the question:** Why did we simplify the inequality $240000x + 300000y \leq 2400000$ by dividing by 60000 in step 7? 2. **Original inequality:**

1. **State the problem:** We need to find (a) the total cost to pay for 1200 bags of produce based on past transport costs, and (b) the number of trips each vehicle (Pickup and Isu

1. **State the problem:** Given the system of equations $$x^3 = 3 - y^3$$ and $$x^6 = 27 - y^6,$$ find the value of $x$. 2. **Recall the formulas and rules:** We know that $$x^6 =

1. **Stating the problem:** Factorise the given expression (though the expression is not provided, we will explain the general method). 2. **Formula and rules:** To factorise an al

1. **State the problem:** Simplify the rational expressions: 5°) \(\frac{a^2 + 4a + 4}{a^2 - 4}\)