Question

the function f is defined by f(x)=7x³. 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)=7/2x³ b) g(x)=7x³ c) g(x)=7x³ + 2 d) g(x)=7x³ - 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) 5 c) 15 d) 45 in the right triangle shown, what is the value of sin a? a) 1/171 b) 35/171 c) 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: Function - shift rule

For a function \(y = f(x)\), shifting it down \(k\) units gives \(y=f(x)-k\). Here \(f(x) = 7x^{3}\) and \(k = 2\), so \(g(x)=7x^{3}-2\).

Step2: Sine - definition in right - triangle

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

Step3: Solve system of equations

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

Step4: Rectangle area formula

The area of a rectangle is \(A=\text{length}\times\text{width}\). Given length \(l = 34\) cm and width \(w = 29\) cm, so \(A=34\times29=986\) \(cm^{2}\).

Answer:

1. D. \(g(x)=7x^{3}-2\)
2. B. \(\frac{35}{171}\)
3. C. \(15\)
4. \(986\) \(cm^{2}\)