Week 7 Complete Flashcards
Linear Functions can be written in 2 ways
Standard form: ππ₯+ππ¦+π=0
Gradient-Intercept form: y=ππ₯+π
Write π¦=3π₯+2 in standard form
π¦=3π₯+2 π¦=3π₯+2(βπ¦) 0=3π₯βπ¦+2 or 3π₯βπ¦+2=0
The gradient intercept form is so called because
if π¦=ππ₯+π then the gradient of the line is π and π is the π¦ value where the line cuts the y axis.
slope is also called the gradient, which is calculated by?
rise/run **(y/x)**
linear function? form
y=mx+c π¦=(3/2)π₯+(1)
dissect linear function form
y=(3/2)x+1 3/2 = gradient/slope = m 1 = y-int = c
y=(3/2)x+1
x intercept (at y = 0)
y=(3/2)x+1
x intercept (at y = 0)
0=(3/2)x+1
- 1 = (3/2)x
- 2/3 = x
y=(3/2)x+1
y intercept (at x = 0)
y=(3/2)x+1
y intercept (at x = 0)
y=(3/2)(0)+1 y=1
Graph 4π₯+2π¦β5=0
using the gradient and y-intercept.
4π₯+2π¦β5=0
2π¦β5 =β4π₯
2y/2= - 4x/2 +5/2
y= -2x + 2 1/2
m = -2 = - 2/1
(means graph will drop from left to right)
y-intercept = 2 1/2
i.e. (0,2 1/2)
Special Linear Equations
There are 2 straight lines that donβt quite fit the form of:
ππ₯+ππ¦+π=0
or
π¦=ππ₯+π.
-
π₯=π
e. g.
π₯=3 (or π₯β3=0)
This means every point on this line, π₯ must be 3.
-
π¦=π
e. g.
π¦=β2(or π¦+2=0)
Likewise, every point on this line π¦ must equal β2.
the line is positive if it rises from the:
and negative if it rises from the:
positive if Left to Right
negative if Right to Left


Another method for plotting the graph:
π¦ = 3/2π₯ + 1

Finding the equation of lines, given one point and the gradient.
Sometimes we know a line passes through a given point and we know the gradient, hence
we can find the equation of this line. This requires the rule:
π - π1 = π(π - π1)
where (π₯1, π¦1) is the point the line passes through and π is the gradient
Find the equation of the line that passes through (-3,4) with a gradient of 2
(-3,4) with a gradient of 2
(π₯1, π¦1) β (-3,4) (i.e. π₯1 = -3, π¦1 = 4)
π = 2
π¦ - π¦1 = π(π₯ - π₯1)
π¦ - 4 = 2(π₯ - (-3))
π¦ - 4 = 2(π₯ + 3)
π¦ - 4 = 2π₯ + 6
π = ππ + ππ
Check: (because (-3,4) lies on this line, when we substitute in π₯ = -3 we should get π¦ = 4)
π¦ = 2 Γ (-3) + 10
π¦ = -6 + 10
π¦ = 4
Also π¦ = 2π₯ + 10 has a gradient of 2
Find equation of lines through 2 points.
If we know 2 points, we can find gradient, then use π¦ - π¦1 = π(π₯ - π₯1)
Example: Find the equation of the line that passes though (-3,4) and (2, -1)
Solution: Although its not necessary, we may find it useful to quickly sketch the 2 points on
the Cartesian Plane. The gradient of the line joining 2 points (π₯1, π¦1) and (π₯2, π¦2)
is
m=y2-y1/x2-x1
(rise/run)

Parallel lines have the same gradient.
π¦ = 3π₯ - 2 π¦ = 3π₯ + 1 6π₯ - 2π¦ + 5 = 0
all have a gradient of 3 and when sketched will be parallel to each other
Find the equation of the line passing though (-4, -2) which is parallel to
3π₯ - 5π¦ + 8 = 0
Solution: We know parallel lines have the same gradient, so we need to find the gradient of
the given line.

π¦ = π1π₯ + π1
π¦ = π2π₯ + π2
product of gradient? perpendicular
π1 Γ π2 = -1
because they are perpendicular, always -1
Find equation of line through (3,4) that is perpendicular to the line
π¦ =2/5 π₯ - 1

If 2 lines are not parallel to each other
then they will intersect each other at one point (π₯, π¦)
only.
Point of Intersection or the Simultaneous
Solution.
There are 2 methods of finding this
unique solution, Substitution and
Elimination.
Solve
3π₯ + 2π¦ - 6 = 0
and
2π₯ - π¦ + 10 = 0
by substitution
The substitution method entails rearranging one of the equations to get either π₯ or π¦ by itself
and then substituting that value into the other equation.
Solution:
3π₯ + 2π¦ - 6 = 0 β(1)
2π₯ - π¦ + 10 = 0 β(2)
Number both equations, and make sure
both are in the form ππ₯ + ππ¦ + π = 0
Upon inspection we can see that in equation (2) it can be easily rearranged to make π¦ the
subject.
(2)β 2π₯ + 10 = π¦ β(2a)
Now substitute for π¦ in equation (1)
(1)β 3π₯ + 2(2π₯ + 10) - 6 = 0
3π₯ + 4π₯ + 20 - 6 = 0
7π₯ + 14 = 0
7π₯ = -14
π₯ = -2
Now substitute for π₯ = -2 in (2a)
2π₯ - 2 + 10 = π¦
6 = π¦
Answer:
π = -π π = **6**
The Elimination method involves
adding or subtracting the equations
Solve
2π₯ + 7π¦ = 16
and
3π₯ - 6π¦ = 2
by elimination
write each equation in standard form
2π₯ + 7π¦ - 16 = 0 β(1)
3π₯ - 6π¦ - 2 = 0 β(2)
We need to make the co-efficients of either π₯ or π¦ the same
(1) Γ 3 6π₯ + 21π¦ - 48 = 0 β(1a)
(2) Γ 2 6π₯ - 12π¦ - 4 = 0 β(2a)
(1a) β (2a) 33π¦ - 44 = 0
33y = 44
y = 44/33 = 4/3
Sub for π¦ = 4/3 in either (1) or (2) to find π₯
in (1) 2x+7*4/3 -16=0
2x+28/3 - 48/3 = 0
2x=20/3
x=10/3
Answer:
x = 10/3
y = 4/3
Solutions to Quadratic Equations
Quadratic functions can be written
π¦=ππ₯2+ππ₯+π
y as a function of x
y = ax2+bx+c
[If π = 0, then π¦ = ππ₯ + π and this is a linear equation or function, so not quadratic.]
Now π¦ is a function of π₯, i.e. π¦ = π(π₯)
Hence a quadratic can also be written π(π₯) = ππ₯2 + ππ₯ + π
Graphing Quadratics:
Letβs graph π¦=π₯2+2π₯β3
This is the shape of the parabola. All quadratics take this shape.
It is symmetrical (a mirror
image) on either side of π₯ = -1
Note that the y-intercept is -3.
Recall from work on linear functions, that with all functions
π₯ intercept is when π¦ = 0
π¦ intercept is when π₯ = 0
π¦ = π₯2 + 2π₯ - 3
π¦ int. (π₯ = 0) π¦ = 02 + 2 Γ 0 - 3
So the π value still gives the π¦ intercept
The π₯ intercept is when π¦ = 0
0 = (π₯ + 3)(π₯ - 1)
Either π₯ = -3 or π₯ = 1
and these are the 2 points where the graph intersects the π₯ axis.
The line of symmetry is π₯ = - π/2π
for π¦ = ππ₯2 + ππ₯ + π
In π¦ = π₯2 + 2π₯ - 3, π = 1, π = 2, π = -3
π₯ = -2/2Γ1
= -2/2
= -1
We can also find the Turning Point of the parabola, i.e. then point where the function has the
greatest or the least value. For this example, it will be the least or minimum value.
Because π₯ = -1 is the line of symmetry, then the least value of the function π¦ = π₯2 + 2π₯ - 3
will occur at π₯ = -1
Substitution π₯ = -1 into this equation gives
π¦ = (-1)2 + 2 Γ (-1) - 3 = 1 - 2 - 3 = -4
So the Turning point is (-1, -4), as can be seen in the previous graph and the minimum
value of the function is -4

Sketch the graph
π¦=β9+6π₯βπ₯2
showing all π₯ and π¦ intercepts. As well state the Domain and Range of this function
Line of symmetry
π₯=βπ/2π
π₯=β6/2Γβ1=3
When π₯=3,
AND π¦=β9+6Γ3β32 = β9+18β9 = 0
So this quadratic only cuts the π₯ axis once at π₯=3.
Note that the parabola is upside down. When βπβ or the coefficient of π₯2 is negative, the parabola is always upside down
Domain of π¦=β9+6π₯βπ₯2
All possible Real π₯ values
ββ<π₯<β
Range: The greatest value π¦ can take is 0, which occurs at the Turning Point.
ββ<π¦β€0

The Discriminant
π2 - 4ππ in the Quadratic Formula is called the Discriminant. By calculating π2 - 4ππ, we can
determine how many times the quadratic function cuts the π₯ axis.

The number (π) of motorists detected speeding on the M1 π days (1 β€ π β€ 10) after the installation of a speed camera can be modelled by the equation: π(π) = π<sup>2</sup>/2 - 3π + 75
Find
a) When least speeding violations occur
b) When most speeding violations occur
π = 1/2, π = -3, π = 75
Line of symmetry
π = - π/2π
π = - -3/2Γ1/2
= 3
π(3) = 32/2 - 3 Γ 3 + 75
π(3) = 4 1/2 - 9 + 75 = 70 1/2
So (3, 70 1/2) is the Turning Point.
Sub in the end values of π, i.e π = 1, π = 10
π = 1 π(1) = 12/2 - 3 Γ 1 + 75
= 1/2 - 3 + 75
= 72 1/2
π = 10 π(10) = 102/2 - 3 Γ 10 + 75
= 50 - 30 + 75
= 95
- *a) Least violations on day 3 (70 1/2)
b) Most violations on day 10 (95)**
A ball is thrown vertically upwards such that its height (metres) above the ground is:
β(π‘) = 40π‘ - 5π‘2
where π‘ is the number of seconds after it is thrown
Find:
a) The maximum height reached
b) The times when it is 15m high
c) The time taken to hit the ground
a) Axis of symmetry π‘ = - π/2π
-40/2Γ-5
=
-40/-10
= 4
At π‘ = 4, Turning Point occurs.
Sub in β(π‘)
β(4) = 40 Γ 4 - 5 Γ 4<sup>2</sup> β(4) = 160 - 80 = 80m
b) β = 15
so 15 = 40π‘ - 5π‘2
5π‘2 - 40π‘ + 15 = 0
Γ· 5 π‘2 - 8π‘ + 3 = 0
π = 1, π = -8, π = 3
π‘ = -πΒ±βπ2-4ππ/2π
π‘ = 8Β±β(-8)2-4Γ1Γ3/2Γ1
π‘ = 8Β±β64-12/2
= 8+β52/2
or
8-β52/2
π‘ = 7.61secs or 0.39secs
c) β = 0
0 = 40π‘ - 5π‘2
0 = 5π‘(8 - π‘)
π‘ = 0 or π‘ = 8
π‘ = 0 corresponds to when the ball is thrown
So π‘ = 8 or 8 seconds before it hits the ground
Linear and Quadratic Functions are examples of functions called?
polynomials.
Linear:
π(π₯) = ππ₯ + π
π β 0
Quadratic:
π(π₯) = ππ₯2 + ππ₯ + π
π β 0
Cubic:
π(π₯) = ππ₯3 + ππ₯2 + ππ₯ + π
π β 0
Quartic:
π(π₯) = ππ₯4 + ππ₯3 + ππ₯2 + ππ₯ + π
π β 0
Graphical Transformation
It is possible to change the shape and position of a function by?
changing the co-efficients of π₯ and the constant term
Linear Transformation
Consider graphs of the form π¦ = ππ₯ + π (π = 0). They will all pass through the origin (0,0)

π = 1 π¦ = π₯
will pass through origin at 45Λ
π = 2 π¦ = 2π₯
Is twice as steep
π = 1/2 π¦ = 1/2x
Is half as steep
and so on
π = -1 π¦ = -π₯
will pass through origin from negative direction (i.e. top
left at 45Λ
π = -2 π¦ = -2π₯
Is twice as steep as π¦ = -π₯
π = -1/2 π¦ = -1/2π₯
Is half as steep
The value of π moves the line vertically. Start with π¦ = π₯

π¦ = π₯ + 3
is parallel to
π¦ = π₯
and moved units 3 up on the y axis
π¦ = π₯ - 2 moves π¦ = π₯ 2 units down
Sketch π¦ = -2π₯ + 3 showing all transformations from π¦ = -π₯
- π¦ = -2π₯
- 3 units up π¦ = -2π₯ + 3

Transformations of Quadratics
Letβs start with a simple quadratic, π¦ = π₯2
π¦ = 2π₯2
is twice as steep
π¦ = 1/2**π₯2
is half as steep
Now
π¦ = -π₯2
and
π¦ = -2π₯2
and
π¦ = -1/2**π₯2

We can move these quadratic graphs vertically
by adding and subtracting values
Adding 2 to π¦ = 2π₯2 moves π¦ = 2π₯2
2 units up the y axis
π¦ = 2π₯2 + 2
Subtracting 3 from π¦ = 2π₯2 gives
π¦ = 2π₯2 - 3 and the graph moves 3
units down the y axis

Show all transformations, starting with π¦ = π₯2,
to sketch π¦ = - 1/3**π₯2 - 2
- π¦ = π₯2
- π¦ = 1/3**π₯2
- π¦ = - 1/3**π₯2
- π¦ = - 1/3**π₯2 - 2

Horizontal Transformations
To move π¦ = π₯2
a) 2 units to right?
b) 3 units to left
a) y = (x - 2)2
b) π¦ = (π₯ + 3)2

The graph π¦ = (π₯ - 1)2 + 4
- π¦ = π₯2
- π¦ = (π₯ - 1)2
- π¦ = (π₯ - 1)2 + 4

Write in standard from
y = 3/5x + 2
y = 3/5x + 2
3/5x - y + 2 = 0
3x - 5y + 10 = 0
Write in gradient-intercept from
3π₯ - 2π¦ + 5 = 0
3π₯ - 2π¦ + 5 = 0
2y = 3x + 5
y = 3/2x + 5/2
Write in gradient-intercept from
π₯ + 3π¦ - 8 = 0
π₯ + 3π¦ - 8 = 0
3y = -x + 8
y = -1/3**x + 8/3
state the gradient and y intercept
2π₯ - 3π¦ - 1 = 0
2x - 3y - 1 = 0
3y = 2x - 1
y = 2/3**x - 1/3
state the gradient and y intercept
π¦ + 3π₯ = 4
y + 3x = 4
y = -3x + 4
state the gradient and y intercept
y = 6
y = 6 y = 0x + 6
Sketch each of these functions showing the π₯ and π¦ intercepts
a) π¦ = 4π₯ - 8
x-intercept is 2, y-intercept is -8, gradient is 4

Sketch each of these functions showing the π₯ and π¦ intercepts
b) 3π₯ - 4π¦ - 2 = 0
3π₯ - 4π¦ - 2 = 0
y = 3x/4 - 1/2
x int 2/3, y int -1/2, gradient 3/4

Sketch each of these functions showing the π₯ and π¦ intercepts
2x + y + 1 = 0
2x + y + 1 = 0
y = -2x - 1
x-intercept is -1/2
y-intercept is -1
gradient is -2

Sketch each of these functions showing the π₯ and π¦ intercepts
π¦ = -3π₯ + 2
x-int 2/3
y-int 2
gradient -3

Find the equation of the line that
a) passes through (-1,4) with gradient 3
y - y1 = m(x - x1)
y - 4 = 3(x + 1)
y - 4 = 3x + 3
y = 3x + 7
Find the equation of the line that
b) passes through (2, -3) with gradient -1
y - y1 = m(x - x1)
y + 3 = -1(x - 2)
y + 3 = -x + 2
y = -x - 1
Find the equation of the line that
c) passes through (-3,1) and (4, -5)
We will start by finding the gradient, m, of the line.
m = y2-y1/x2-x1
m = -5-1/4+3
m = -6/7
Now we can find the equation of the line that passes through the two points.
y - y1 = m(x - x1)
y -1 = -6/7(x + 3)
y -1 = -6/7**x - 18/7
y = -6/7**x - 11/7
Find the equation of the line that
d) passes through (-4,0) and (0,3)
start by finding the gradient of the line
m = y2-y1/x2-x1
= 3-0/0+4
= 3/4
Now we can find the equation of the line that passes through the two points.
y - y1 = m(x - x1)
y - 0 = 3/4(x + 4)
y = 3/4**x + 3
Find the equation of the line that
e) passes through (1,8) and parallel to
π¦ = 2/3**π₯ + 4
If the line is parallel to y = 2/3**x + 4, then the gradient of the line will be 2/3
y - y1 = m(x - x1)
y - 8 = 2/3**(x - 1)
y - 8 = 2/3**x - 2/3
y = 2/3**x + 22/3
Find the equation of the line that
f) passes through (-2, -4) and parallel to 5π₯ - π¦ + 1 = 0
We will start by finding the gradient of 5x - y + 1 = 0
5x - y + 1 = 0
y = 5x + 1
Therefore, the gradient is 5.
y - y1 = m(x - x1)
y + 4 = 5(x +2)
y +4 = 5x + 10
y = 5x + 6
g) passes through (5,1) and perpendicular to 3π₯ + 2π¦ = 0
We will start by finding the gradient of 3x + 2y = 0
3x + 2y = 0
2y = -3x
y = -3/2**x
Therefore, as the line we are trying to find it perpendicular to 3x+2y = 0, the gradient of the line we are trying to find must be 2/3.
(Remember the gradients of perpendicular
lines multiply to give -1).
y-y1=m(x-x1)
y - 1 = 2/3**(x-5)
y - 1 = 2/3**x - 10/3
y = 2/3**x - 7/3
Solve these simultaneous equations (then check your answers)
a)
-2π₯ + 7π¦ = 8
π¦ + 1 = 3π₯
-2x + 7y = 8 (1)
y + 1 = 3x (2)
Rearranging the second equation gives
-2x +7y = 8 (1)
3x -y = 1 (2)
To eliminate the y term, we will calculate (1) + 7 Γ (2).
19x = 15
x = 15/19
To find y, we will substitute x = 15/19 into Equation (2). (It doesnβt matter which equation is used to solve for y. Equation (2) was chosen as it appeared to be the
easiest to use to solve for y).
y + 1 = 3 Γ 15/19
y + 1 = 45/19
y = 26/19
Thus,
x = 15/19
y = 26/19
Solve these simultaneous equations (then check your answers)
b)
2π₯ - π¦ - 11 = 0
4π₯ + π¦ - 8 = 0
2x - y - 11 = 0 (1)
4x + y - 8 = 0 (2)
Simply adding Equations (1) and (2) will eliminate the y term
6x - 19 = 0
6x = 19
x = 19/6
to find y we will substitute x = 19/6 into Equation (1)
2 * 19/6 - y - 11 = 0
-y - 14/3 = 0
y = -14/3
Thus,
x = 19/6
y = -14/3
Solve these simultaneous equations (then check your answers)
c)
-3π₯ + π¦ - 8 = 0
4π₯ + 2π¦ - 21 = 0
-3x + y - 8 = 0 (1)
4x + 2y - 21 = 0 (2)
To eliminate the y term, we will calculate 2 Γ (1) - (2).
-10x + 5 = 0
10x = 5
x = 1/2
To find y, we will substitute x = 1/2 into Equation (1)
-3 Γ 1/2 + y - 8 = 0
y - 19/2 = 0
y =19/2
Therefore,
x = 1/2 y = 19/2
Solve these simultaneous equations (then check your answers)
d)
4π₯ - 7π¦ = 21
6 = -2π₯ + π¦
4x - 7y = 21 (1)
-2x + y = 6 (2)
To eliminate the x term, we will calculate (1) + 2 Γ (2)
-5y = 33
y = -33/5
To find x, we will substitute y = -33/5 into Equation (1).
4x - 7 * -33/5 = 21
4x + 231/5 = 21
4x = -126/5
x = -63/10
thus
x = -63/10
y = -33/5
a) fill in the table
b) Plot the 9 ordered pairs (π₯, π¦) on the Cartesian plane
c) Draw a smooth curve through all 9 points
d) Calculate and draw in the line of symmetry
e) Label all π₯ and π¦ intercepts
f) Calculate and label the co-ordinates of the Turing Point.
g) State the Domain and Range of π(π₯) = -π₯2 + 3π₯ + 4
y = -x2 + 3x + 4
x -4 -3 -2 -1 0 1 2 3 4
y -24 -14 -6 0 4 6 6 4 0
(b) See graph below.
(c) See graph
(d) The line of symmetry is at x = 1.5. It is shown on the graph using a red, dashed line.
(e) See graph
(f) There is a turning point, a maximum, at (1.5, 6.25).
(g) Domain: All real values of x, Range: y β€ 6.25
(a)

Use the Discriminant to determine the number of intercepts on the π₯ axis
a) π¦ = 3π₯2 + 4π₯ - 1
To find the discriminant, D, of the quadratic y = ax2 + bx + c, calculate D = b2 - 4ac.
If the discriminant is positive (D>0), the quadratic has two x-intercepts.
If the discriminant is zero (D = 0), the quadratic only has one x-intercept.
If the discriminant is negative (D<0), the quadratic has no x-intercepts.
y = 3x<sup>2</sup> + 4x - 1 D = 4<sup>2</sup> - 4 Γ 3 Γ -1 = 28
As the discriminant is positive, the quadratic has two x-intercepts.
Use the Discriminant to determine the number of intercepts on the π₯ axis
b)
π¦ = π₯2 + 4π₯ + 4
y = x<sup>2</sup> + 4x + 4 D = 4<sup>2</sup> - 4 Γ 1 Γ 4 = 0
As the discriminant is zero, the quadratic has one x-intercept.
Use the Discriminant to determine the number of intercepts on the π₯ axis
c)
π¦ = -π₯2 - 10π₯ + 2
y = -x2 - 10x + 2
D = (-10)2 - 4 Γ -1 Γ 2 = 108
As the discriminant is positive, the quadratic has two x-intercepts.
Use the Discriminant to determine the number of intercepts on the π₯ axis
d)
π¦ = -2π₯2 + π₯ - 1
y = -2x2 + x - 1
D = 12 - 4 Γ -2 Γ -1 = -7
As the discriminant is negative, the quadratic has no x-intercepts.
Show all transformations in a sketch
a) Start with π¦ = -π₯ Finish with π¦ = -2π₯ + 3
- y = -x is drawn in blue
- y = -2x is drawn in red
- y = -2x + 3 is drawn in black
To draw a graph of y = -2x+3, it may be easier to just start with a graph of y = -2x.

Show all transformations in a sketch
b) Start with π¦ = π₯2 Finish with π¦ = 2π₯2 - 4
- y = x2 is drawn in blue
- y = 2x2 is drawn in red
β’ y = 2x2 - 4 is drawn in black
To draw a graph of y = 2x2 -4, it may be easier to just start with a graph of y = 2x2

Show all transformations in a sketch
c) Start with π¦ = π₯2 Finish with π¦ = -1/4**π₯2 + 2
- y = x2 is drawn in blue
- y = -1/4**x2 is drawn in red
- y = -1/4**x2 + 2 is drawn in black

Show all transformations in a sketch
d) Start with π¦ = π₯2 Finish with π¦ = (π₯ - 3)2 + 1
- y = x2 is drawn in blue
- y = (x - 3)2 is drawn in red
- y = (x - 3)2 + 1 is drawn in black

The volume of water in a tank, π(π‘) m3, over a 10 month period is given by the function
π(π‘) = 2π‘2 - 16π‘ + 40 where π‘ is in months and π‘ β [0,10]
Find a) The volume when π‘ = 0
V (0) = 2 Γ 02 - 16 Γ 0 + 40
= 40
Initially (t = 0) there is 40 m3 of water in the tank
The volume of water in a tank, π(π‘) m3, over a 10 month period is given by the function
π(π‘) = 2π‘2 - 16π‘ + 40 where π‘ is in months and π‘ β [0,10]
b) The volume when π‘ = 10
V (10) = 2 Γ 102 - 16 Γ 10 + 40
= 80
After 10 months (t = 10) there is 80 m3 of water in the tank.
The volume of water in a tank, π(π‘) m3, over a 10 month period is given by the function
π(π‘) = 2π‘2 - 16π‘ + 40 where π‘ is in months and π‘ β [0,10]
c) The minimum volume over the 10 months
To find the minimum volume in the tank over the first 10 months, we will start by drawing a graph of V (t).
From the graph, it can be seen that the minimum volume in the tank occurs at t = 4.
V (4) = 2 Γ 42 - 16 Γ 4 + 40
= 8
Alternatively, if we didnβt want to draw a graph, the x value of the turning point is given by
x = -b/2a
= 16/2*2
= 4
Thus the minimum volume in the tank over the 10 month period is 8 m3

The volume of water in a tank, π(π‘) m3, over a 10 month period is given by the function
π(π‘) = 2π‘2 - 16π‘ + 40 where π‘ is in months and π‘ β [0,10]
d) The maximum volume over the 10 months
To find the maximum volume in the tank over the 10 month period, we need to test the end points, t = 0 and t = 10.
The maximum volume will not occur at the turning
point as it was a minimum.
V (0) = 40 and V (10) = 80 (found in part (a)).
Therefore
the maximum volume of water in the tank over the 10 month period is 80 m3
The parabolic Arch of a bridge spans a river 120m wide. The highest point of the arch is
30m above the water. Calculate a mathematical model β(π) that describes the height of the
arch above the water at any point π metres across the river.
The graph is a little hard to see, but we have the following information about the function h(d) where h is the height of the arch, in metres, above the water and d is the distance, in metres, across the river.
h(0) = 0 h(60) = 30 h(120) = 0
We have also been told that h(d) is a parabolic function (a quadratic). Let h(d) = ad<sup>2</sup> + bd + c
where a, b and c are constants that we need to determine. Using h(0) = 0 gives
h(0) = a Γ 02 + b Γ 0 + c = 0
c = 0
Using h(120) = 0 gives
h(120) = a Γ 1202 + b Γ 120 = 0
b Γ 120 = -a Γ 1202
b = -120a
Using h(60) = 30 and b = -120a gives h(60) = a Γ 60<sup>2</sup> + b Γ 60 = 30 3600a + 60 Γ -120a = 30 3600a - 7200a = 30 -3600a = 30 a = -1/120
Finally, using b = -120a and that a = -1/120 gives
b = -120 * -1/120 = 1
Therefore, putting all the information together, the function h(d) is given by
h(d) = ad2 + bd + c
= -1/120 Γ d2 + 1 Γ d + 0
= -d2/120 + d