Question

HELP PLEASE!!! 25 Points & Brainliest!

Answer 1

Answer:  The answers are:

(1) $g(x)=3x^2+12x+16.$

(2) $f\left(\dfrac{1}{2}\right)=\dfrac{5}{3}.$

(3) (a) $y+9=2(x+1).$

Step-by-step explanation:  The calculations are as follows:

(1) The given function is

$f(x)=3x+4.$

We are to find the equation for the function g(x) is it represents f(x) after a horizontal translation of 2 units to the left.

When the function f(x) is translated 2 units to the left, the vertex of the function will shift from (0, 4) to (-2, 4).

So, the equation of g(x) will be

$g(x)=3(x-(-2))^2+4\\\\\Rightarrow g(x)=3(x+2)^2+4\\\\\Rightarrow g(x)=3x^2+12x+12+4\\\\\Rightarrow g(x)=3x^2+12x+16.$

Thus, the equation for g(x) is

$g(x)=3x^2+12x+16.$

(2)  The given function is

$f(x)=-\dfrac{2}{3}x+2.$

We are to find the value of $f\left(\dfrac{1}{2}\right).$

We have, after putting $x=\dfrac{1}{2}$ in the definition of f(x) that

$f\left(\dfrac{1}{2}\right)=-\dfrac{2}{3}\times \dfrac{1}{2}+2=-\dfrac{1}{3}+2=\dfrac{5}{3}.$

Thus,

$f\left(\dfrac{1}{2}\right)=\dfrac{5}{3}.$

(3) We are to find the equation of a line in point-slope form for the following point and slope:

point: (1,−9) ; slope: 2.

We know that the equation of a line in point-slope form passing through the point (a, b) and slope m is given by

$y-b=m(x-a).$

Therefore, the equation for the given line will be

$y-(-9)=2(x-1)\\\\\Rightarrow y+9=2(x+1).$

Thus, the required equation of the line in point-slope form is

$y+9=2(x+1).$

Option (a) is CORRECT.

Hence, all the questions are answered.

Answer 2

Answer:

(1) $f(x)=3x+4$

Ad g(x) represent f(x) after horizontal translation 2 units to the left.

$g(x)=3(x+2)+4$

$g(x)=3x+6+4$

$g(x)=3x+10$ is the required translation.

(2) $f(x)=\frac{-2x}{3}+2$

$f(\frac{1}{2})=\frac{-2}{3}\cdot(\frac{1}{2})+2$

$f(\frac{1}{2})=\frac{5}{3}$

(3) We have been given a point (1,-9); slope 2

We will use the equation:

$y-y_1=m(x-x_1)$

m is the give slope.

$y-(-9)=2(x-1)$

$y+9=2(x-1)$

Therefore, Option (a) is correct.