Answer:
see explanation
Step-by-step explanation:
(43)
To find f(- 12) substitute x = - 12 into f(x)
f(- 12) = [tex]\frac{1}{3}[/tex](- 12) + 8 = - 4 + 8 = 4, thus
3f(- 12) + 4 = 3(4) + 4 = 12 + 4 = 16
(44)
Given
f(x) = 2x² + 3x - 4 , then
f(x + 1) = 2(x + 1)² + 3(x + 1) - 4 ← expand factors
= 2(x² + 2x + 1) + 3x + 3 - 4
= 2x² + 4x + 2 + 3x - 1
= 2x² + 7x + 1
-----------------------------------------
f(- x) = 2(- x)² + 3(- x) - 4 = 2x² - 3x - 4
(45)
The equation of a line in slope intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Given
4x + 6y + 5 = 0 ( subtract 4x + 5 from both sides )
6y = - 4x - 5 ( divide all terms by 6 )
y = - [tex]\frac{2}{3}[/tex] x - [tex]\frac{5}{6}[/tex] ← in slope- intercept form
with slope m = - [tex]\frac{2}{3}[/tex]
Parallel lines have equal slopes, thus
y = - [tex]\frac{2}{3}[/tex] x + c ← is the partial equation
To find c substitute (5, 2) into the partial equation
2 = - [tex]\frac{10}{3}[/tex] + c ⇒ c = 2 +
y = - [tex]\frac{2}{3}[/tex] x + [tex]\frac{16}{3}[/tex] ( multiply through by 3 )
3y = - 2x + 16 ( subtract - 2x + 16 from both sides )
2x + 3y - 16 = 0 ← equation of parallel line
----------------------------------------------------------------
Given a line with slope m then the equation of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-\frac{2}{3} }[/tex] = [tex]\frac{3}{2}[/tex] , thus
y = [tex]\frac{3}{2}[/tex] x + c ← is the partial equation
To find c substitute (5, 2) into the partial equation
2 = [tex]\frac{15}{2}[/tex] + c ⇒ c = 2 -
y = [tex]\frac{3}{2}[/tex] x - [tex]\frac{11}{2}[/tex] ( multiply through by 2 )
2y = 3x - 11 ( subtract 2y from both sides )
3x - 2y - 11 = 0 ← equation of perpendicular line
