🧮 algebra

1. The problem is to understand the transformation of a function when its output changes from 2 to 4. 2. A vertical stretch by a scale factor $k$ means every $y$-value of the funct

1. **State the problem:** We need to find how many adult tickets were sold given that a total of 804 tickets were sold, consisting of adult and student tickets, and there were 54 m

1. The problem asks to describe a sequence of transformations that transforms the graph of $y = g^{-1}(x)$ to the graph of $y = f(x)$. 2. From part (b)(i), we know $g^{-1}(x) = 2^{

1. **State the problem:** We need to find the equation of the logarithmic function $f(x)$ given its graph. 2. **Identify key features:** The graph has a vertical asymptote at $x=4$

1. The problem asks to factor the given polynomial expressions. 2. We use the formula for the cube of a binomial: $$ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 $$

1. The problem is to analyze the quadratic function $$y = ax^2 + bx + c$$ and understand how changing the coefficients $a$, $b$, and $c$ affects the graph and the discriminant $$D

1. **Stating the problem:** We want to understand how changing the coefficients $a$, $b$, and $c$ in the quadratic equation $y = ax^2 + bx + c$ affects the discriminant $D = b^2 -

1. **State the problem:** We need to find the value of $a$ in the equation $$a = \frac{b^2 - 6}{c^2 - 2}$$ when $b = 4$ and $c = 2$. 2. **Write the formula:** The formula given is

1. **Problemstellung:** Schreibe die Terme ohne Malpunkt, wenn möglich. 2. **Formel und Regeln:** Der Malpunkt \(\cdot\) kann weggelassen werden, wenn eine Zahl direkt vor einer Va

1. **State the problem:** Complete the prime factor trees for the given numbers. 2. **Recall the prime factorization rule:** Every composite number can be expressed as a product of

1. **Problem:** Solve for $x$ in the equation $x = \log_3 17$, correct to 3 decimal places. 2. **Formula:** The change of base formula for logarithms is

1. Il problema chiede di trovare due numeri la cui somma è 16 e la somma dei loro quadrati è 130. 2. Indichiamo i due numeri con $x$ e $y$.

1. **Stating the problem:** Trova due frazioni la cui somma è $\frac{5}{6}$ e il cui prodotto è $\frac{1}{6}$.

1. **State the problem:** Solve the system of equations: $$ (5x)^2 + (5y)^2 = 148 $$

1. The problem asks us to find the equation of a function $h(x)$ that first stretches the graph of $f(x) = |x|$ vertically by a factor of 4, then shifts it down by 2 units. 2. The

1. The problem asks to find the equation of a function $h(x)$ that first reflects the graph of $f(x) = |x|$ over the y-axis, then shifts it down 3 units. 2. Reflection over the y-a

1. The problem is to find the formula for the gradient (slope) of a line given two points or a function. 2. The gradient formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is

1. Das Problem lautet: Berechne den Wert von $y$. Da keine weitere Information gegeben ist, ist die Aufgabe unvollständig. 2. Um $y$ zu berechnen, benötigen wir eine Gleichung oder

1. **Problemstellung:** Überprüfe, ob der Punkt $P(1,3)$ auf der Parabel $y = (x - 3)^2 - 1$ liegt. 2. **Formel:** Die Parabel ist definiert durch die Gleichung $$y = (x - 3)^2 - 1

1. **Write the augmented matrix for the system:** Given the system:

1. The problem is to convert the given augmented matrix $$\begin{bmatrix}4 & -2 & -1 & 1 \\ -3 & 0 & -4 & -7\end{bmatrix}$$