units 1 and 2 methods revision

Find the equation of the line that passes through the point (-3, 1) and is parallel to the line 4y - 5x + 1 = 0.

$4y-5x+1=0$
$y=\frac{5}{4}x-\frac{1}{4}$
$m=\frac{5}{4}$
$y-y_1=m(x-x_1)$

$y-1=\frac{5}{4}x+\frac{15}{4}$
$y=\frac{5}{4}x+\frac{19}{4}$

Find the equation of the line that passes through the midpoint of (-2, 4) and (6, 10) with a gradient of $\frac{-1}{3}$.

The midpoint is $(\frac{-2+6}{2},\frac{4+10}{2})$ which is $(2,7)$
$y-y_1=m(x-x_1)$

$y-7=\frac{-1}{3}(x-2)$
$3y+x-23=0$

Two containers contain a total of 800 litres between them. The liquid flows from one container to another. One container holds 200 litres less than three times the contents of the other container. Set up a set of simultaneous equations to model this situation. Find the amount in each container.

Let the volume of the containers be x and y respectively. We know $y=3x-200$
$x+y=800$
$x+3x-200=800$
$4x=1000$
$x=250$

$y=550$

Find the values) of y so that the distance between the points (5, y) and (8, -1) is 5 units.

The distance is $\sqrt{(5-8{)}^2+(y+1{)}^2}=5$
$(5-8{)}^2+(y+1{)}^2=25$
$9+(y+1{)}^2=25$
$(y+1{)}^2=16$
$y+1=\pm 4$
$y=\pm 4-1$
$y=3,-5$

A line passes through the midpoint of the interval AB where A is the point (-3, 11) and B is the point (7, -7) such that it makes an angle of 135° with the positive direction of the x-axis.

Find the midpoint of AB.

$(\frac{-3+7}{2},\frac{11-7}{2})=(2,2)$

Find the gradient of the line.

$\tan (135°)=m=-1$

Find the equation of the line.

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

$y-2=-x+2$
$y=-x+4$

The range of the function f. x ≤ 1 → R where f(x) = 5 - 2x is:

$f(1)=3$
gradient is negative
range is $[3,\infty )$

Find the turning point of the graph of $y=3x^2-12x+17$.

TP at $x=\frac{-b}{2a}=\frac{12}{6}=2$
$y(2)=5$
TP is $(2,5)$

State the domain and range of $y=15-3(7-x{)}^3,x\le -2$.

We are given the domain.
$y(-2)=-2172$

$y=15-3(-x+7{)}^3$
Have a rough sketch. This is a positive cubic, a one to one function.
Therefore the range is $(-\infty ,-2172)$

State the domain and range of $y=-2(x-2{)}^4+2)+3,x>-2$.