Question

8 the value of a piece of machinery, f(x), depends on the age of the machinery in years, x. according to the function f(x)=8000(0.7)^x. which graph models this relationship? 9 which function is equivalent to f(x)=3x^3 - x^2 - 3x + 1? a. f(x)=(x^2 + 3x - 1)(3x - 1) b. f(x)=(3x^2 - 2x - 1)(x + 1) c. f(x)=(3x^2 + 2x - 1)(x - 1) d. f(x)=(x^2 - 2x - 1)(3x + 1) 10 which is a characteristic of any function created by the product of two linear factors? a. has a maximum b. has two x - intercepts c. has two y - intercepts d. has an axis of symmetry

Explanation:

8.

Step1: Identify function type

The function $f(x)=8000(0.7)^x$ is an exponential - decay function. The general form of an exponential function is $y = a\cdot b^x$, where $a = 8000$ and $b=0.7$. Since $0 < b<1$, as $x$ increases, $y$ decreases.

Step2: Analyze graphs

• Graph a is an increasing linear function.
• Graph b is a parabola (quadratic function).
• Graph c is a decreasing exponential - decay function.
• Graph d is a cubic function.

Step1: Expand each option

• Option a: $(x^{2}+3x - 1)(3x - 1)=3x^{3}-x^{2}+9x^{2}-3x-3x + 1=3x^{3}+8x^{2}-6x + 1$.
• Option b: $(3x^{2}-2x - 1)(x + 1)=3x^{3}+3x^{2}-2x^{2}-2x-x - 1=3x^{3}+x^{2}-3x - 1$.
• Option c: $(3x^{2}+2x - 1)(x - 1)=3x^{3}-3x^{2}+2x^{2}-2x-x + 1=3x^{3}-x^{2}-3x + 1$.
• Option d: $(x^{2}-2x - 1)(3x + 1)=3x^{3}+x^{2}-6x^{2}-2x-3x - 1=3x^{3}-5x^{2}-5x - 1$.

Step1: Recall the form of a function from two linear factors

If a function $y=(ax + b)(cx + d)=acx^{2}+(ad+bc)x+bd$ (a quadratic function), the $x$ - intercepts are found by setting $y = 0$. So, $(ax + b)(cx + d)=0$, which gives $x=-\frac{b}{a}$ and $x =-\frac{d}{c}$. A quadratic function has two $x$ - intercepts (unless the two linear factors are the same, in which case it has a repeated root).

• A quadratic function may have a maximum or a minimum, not always a maximum.
• A function has only one $y$ - intercept (found by setting $x = 0$).
• Only quadratic functions (from two non - parallel linear factors) have an axis of symmetry, but the question asks for a general characteristic of a function from two linear factors.

Answer:

c

9.