starxox
contestada

Use the functions f(x) and g(x) to determine which function has the smallest zero and provide its coordinates.

f(x) = 3x2 + 18x − 21


x g(x)
18 −17
19 0
20 19
21 40
22 63
g(x); (19, 0)
g(x); (−17, 0)
f(x); (−7, 0)
f(x); (1, 0)

Respuesta :

danielmaduroh

Answer:

[tex]f(x); (-7, 0)[/tex]

Step-by-step explanation:

The zeros of a function are those values where the graph of the function touches the x-axis. First, we have [tex]f(x)[/tex] which is a parabola defined by the following equation:

[tex]f(x)=3x^2+18x-21[/tex]

By using the quadratic formula, we can get the zeros, therefore:

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\ a=3, \ b=18, \ c=-21 \\ \\ \\ x=\frac{-18\pm \sqrt{(18)^2-4(3)(-21)}}{2(3)} \\ \\ x=\frac{-18\pm \sqrt{324+252}}{6} \\ \\ x_{1}=1 \ and \ x_{2}=-7[/tex]

So the zeros of [tex]f(x)[/tex] are [tex]x_{1}=1 \ and \ x_{2}=-7[/tex]. The zero of [tex]g(x)[/tex] is just one and can be determined from the table, which is [tex]x=19[/tex]. So we can see that [tex]f(x)[/tex] is the function that has the smallest zero, which is:

[tex]\boxed{f(x); \ (-7, 0)}[/tex]

Answer:

The smallest zero is (-7,0) and correspond to f(x)

Step-by-step explanation:

x g(x)    difference of g(x)   difference of differences

18 −17

19 0         0 - (-17) = 17                          

20 19        19  - 0 = 19                  19 - 17 = 2

21 40       40 - 19 =21                 21 - 19 = 2

22 63       63 - 40 =23              23 - 21 = 2

Then, g(x) is a quadratic function. The regression gives: g(x) = x^2 - 20x +19 (I made it in Excel, you can use any similar software or a calculator).

Using the quadratic formula, the zeros of g(x) are:

[tex]x = \frac{20 \pm \sqrt{(-20)^2 - 4(1)(19)}}{2(1)} [/tex]

[tex]x = \frac{20 \pm 18}{2} [/tex]

[tex]x_1 = \frac{20 + 18}{2} [/tex]

[tex]x_1 = 19 [/tex]

[tex]x_2 = \frac{20 - 18}{2} [/tex]

[tex]x_2 = 1 [/tex]

Coordinate of the zeros: (19, 0) and (1,0)

Using the quadratic formula, the zeros of f(x) are:

[tex]x = \frac{-18 \pm \sqrt{18^2 - 4(3)(-21)}}{2(3)} [/tex]

[tex]x = \frac{-18 \pm 24}{6} [/tex]

[tex]x_1 = \frac{-18 + 24}{6} [/tex]

[tex]x_1 = 1 [/tex]

[tex]x_2 = \frac{-18 - 24}{6} [/tex]

[tex]x_2 = -7[/tex]

Coordinate of the zeros: (-7, 0) and (1,0)

Otras preguntas