Question

the function f is defined by f(x)=2x². in the xy - plane, the graph of y = g(x) is the result of shifting the graph of y = f(x) down 2 units. which equation defines function g? a) g(x)=\frac{2}{3}x² b) g(x)=2x² c) g(x)=2x² + 2 d) g(x)=2x²-2. y=-3x 4x + y = 15. the solution to the given system of equations is (x,y). what is the value of x? a) 1 b) 3 c) 15 d) 4. in the right - triangle shown, what is the value of sin a? a) \frac{1}{171} b) \frac{35}{171} c) \frac{171}{35} d) 171. what is the area, in square centimeters, of a rectangle with a length of 34 centimeters (cm) and a width of 29 cm?

Explanation:

Step1: Recall function - translation rule

When the graph of $y = f(x)$ is shifted down $k$ units, the new function is $y=f(x)-k$. Here $f(x) = 7x^{2}$ and $k = 3$, so $g(x)=7x^{2}-3$.

Step2: Recall sine - definition in a right - triangle

In a right - triangle, $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$. For angle $A$, the opposite side to angle $A$ is $35$ and the hypotenuse is $171$, so $\sin A=\frac{35}{171}$.

Step3: Solve the system of equations

We have the system

$$\begin{cases}y=-3x\\4x + y=15\end{cases}$$

. Substitute $y=-3x$ into $4x + y=15$, we get $4x-3x=15$, which simplifies to $x = 15$.

Step4: Calculate the area of a rectangle

The area of a rectangle is given by $A = lw$, where $l = 36$ cm and $w = 29$ cm. So $A=36\times29=1044$ $cm^{2}$.

Answer:

1. D. $g(x)=7x^{2}-3$
2. B. $\frac{35}{171}$
3. C. $15$
4. $1044$