Answer:
(1) [tex]f(x)=3x+4[/tex]
Ad g(x) represent f(x) after horizontal translation 2 units to the left.
[tex]g(x)=3(x+2)+4[/tex]
[tex]g(x)=3x+6+4[/tex]
[tex]g(x)=3x+10[/tex] is the required translation.
(2) [tex]f(x)=\frac{-2x}{3}+2[/tex]
[tex]f(\frac{1}{2})=\frac{-2}{3}\cdot(\frac{1}{2})+2[/tex]
[tex]f(\frac{1}{2})=\frac{5}{3}[/tex]
(3) We have been given a point (1,-9); slope 2
We will use the equation:
[tex]y-y_1=m(x-x_1)[/tex]
m is the give slope.
[tex]y-(-9)=2(x-1)[/tex]
[tex]y+9=2(x-1)[/tex]
Therefore, Option (a) is correct.
Answer: The answers are:
(1) [tex]g(x)=3x^2+12x+16.[/tex]
(2) [tex]f\left(\dfrac{1}{2}\right)=\dfrac{5}{3}.[/tex]
(3) (a) [tex]y+9=2(x+1).[/tex]
Step-by-step explanation: The calculations are as follows:
(1) The given function is
[tex]f(x)=3x+4.[/tex]
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
[tex]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.[/tex]
Thus, the equation for g(x) is
[tex]g(x)=3x^2+12x+16.[/tex]
(2) The given function is
[tex]f(x)=-\dfrac{2}{3}x+2.[/tex]
We are to find the value of [tex]f\left(\dfrac{1}{2}\right).[/tex]
We have, after putting [tex]x=\dfrac{1}{2}[/tex] in the definition of f(x) that
[tex]f\left(\dfrac{1}{2}\right)=-\dfrac{2}{3}\times \dfrac{1}{2}+2=-\dfrac{1}{3}+2=\dfrac{5}{3}.[/tex]
Thus,
[tex]f\left(\dfrac{1}{2}\right)=\dfrac{5}{3}.[/tex]
(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
[tex]y-b=m(x-a).[/tex]
Therefore, the equation for the given line will be
[tex]y-(-9)=2(x-1)\\\\\Rightarrow y+9=2(x+1).[/tex]
Thus, the required equation of the line in point-slope form is
[tex]y+9=2(x+1).[/tex]
Option (a) is CORRECT.
Hence, all the questions are answered.
