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Quadratic Functions
FUNDAMENTAL COACHING
CENTRE( FCC)
Quadratic functions
A relation defined by the rule
(where a ≠ 0 and a, b and c are constants) is called a quadratic
function.
e.g.
In this section, we will learn the general characteristics of
quadratic functions by plotting their graphs. Then these
characteristics will be used to sketch other quadratic graphs.
Graphs of
Example 1:
Plot the graph of
for -3 ≤ x ≤ 3
x -3 -1 0
y 9 1 0
1
1
3
9
The parabola opens upwards. The
minimum value of y is zero and it
occurs when x = 0. The point
(0, 0) is called the turning point or
vertex of the parabola.
When we plot these points and join them with a smooth
curve, we obtain the quadratic graph shown above. The
curve is called a parabola. It has many applications in science
and engineering. For example, the path followed by a
projectile or the shape of the reflector in a car's headlamps or
searchlights.
Looking at the graph and the shape of the curve, you
could imagine that a mirror is placed along the y-axis: the lefthand side and right hand side of the curve are mirror-images
of each other. This property is called symmetry. We say that
the graph is symmetrical about the y-axis, and the y-axis is
called the axis of symmetry. So, the axis of symmetry has
equation x = 0.
Example 2:
Plot the graph of
for -2 ≤ x ≤ 2
x
y
-2 -1 0
2 0.5 0
1 2
0.5 2
The graph is a parabola which opens
upwards. The minimum value of y is 0
and it occurs
when x = 0. The point (0, 0) is called
the vertex of the parabola. The graph is
symmetrical
about x = 0, i.e. the y-axis.
Graphs of
Example 3:
Plot the graph of
for -3 ≤ x ≤ 3
The graph is a parabola which opens
downwards. Clearly, the graph is
symmetrical about the y-axis.
So, the equation of the axis of symmetry is
x = 0.
The maximum value of y is 0 and it
occurs when x = 0.
The vertex (or turning point) of the
parabola is the point (0, 0).
x -3 -1 0 1 3
y -9 -1 0 -1 -9
Example 4:
Plot the graph of
for -2 ≤ x ≤ 2
The graph is a parabola which opens
downwards.
Clearly, the graph is symmetrical about
the y-axis. So, the equation of the axis of
symmetry is x = 0.
The maximum value of y is 0 and it
occurs when x = 0.
The vertex of the parabola is the point (0,
0).
x
y
-2
-8
-1
-2
0
0
1
-2
2
-8
Graphs of
Example 5:
Plot the graph of
for -2 ≤ x ≤ 2
The graph is a parabola which opens
upwards. Clearly, the graph is
symmetrical about the y-axis. So,
the equation of the axis of symmetry
is x = 0.
The minimum value of y is 3 and it
occurs when x = 0.
The vertex of the parabola is the
point (0, 3).
x
y
-2
7
-1
4
0
3
1
4
2
7
Graphs of
Example 6:
Plot the graph of
for -3 ≤ x ≤ 3
The graph is a parabola which opens
downwards. Clearly, the graph is
symmetrical about the y-axis. So, the
equation of the axis of symmetry is x =
0.
The maximum value of y is 1 and it
occurs when x = 0.
The vertex of the parabola is the point
(0, 1).
x
y
-3
-8
-1
0
0
1
1
0
3
-8
Graphs of
Example 7:
Plot the graph of
for 0 ≤ x ≤ 4
The graph is a parabola which opens
upwards.
The graph is symmetrical about the line x
= 2. So, the equation of the axis of
symmetry is x = 2.
The minimum value of y is 0 and it occurs
when x = 2.
The vertex of the parabola is (2, 0).
x
y
0
4
1
1
2
0
3
1
4
4
Graphs of
Example 8:
Plot the graph of
for 0 ≤ x ≤ 4
The graph is a parabola which opens
downwards.
The graph is symmetrical about the
line x = 2. So, the equation of the axis
of symmetry is x = 2.
The maximum value of y is 0 and it
occurs when x = 2.
The vertex of the parabola is (2, 0).
x
y
0
-4
1
-1
2
0
3
-1
4
-4
Graphs of
Example 9:
Plot the graph of
for -1 ≤ x ≤ -3
The graph is a parabola which opens
upwards.
The graph is symmetrical about the line
x = 1. So, the equation of the axis of
symmetry is x = 1.
The minimum value of y is 2 and it
occurs when x = 1.
The vertex of the parabola is (1, 2).
x
y
-1
6
0
3
1
2
2
3
3
6
Graphs of
Example 10:
Plot the graph of
for -1 ≤ x ≤ -3
The graph is a parabola which opens
downwards.
The graph is symmetrical about the line
x = 1. So, the equation of the axis of
symmetry is x = 1.
The maximum value of y is 2 and it
occurs when x = 1.
The vertex of the parabola is (1, 2).
x
y
-1
-2
0
1
1
2
2
1
3
-2
Quadratic graphs by transformations
It is useful to consider some graphs as the dilation,
translation or reflection of a basic graph.
Dilation and Reflection
Dilation has the effect of stretching or compressing a
graph. Dilation is either from the x-axis or the
y-axis. Also, it could be from both axes.
Consider the graphs of the following functions on the same set
of axes:
We notice that:
Translation
A translation is a shift of a graph in a given direction.
Consider the graphs of the following functions on the same set
of axes:
Graphs of y = (x - b)²
Consider the graphs of the following functions on the same set
of axes:
Example 11:
Sketch the graph of
Solution:
Sketching Parabolic Graphs
•
•
•
•
turning point
axis of symmetry
x-intercepts
y-intercept
Example 12:
Sketch the graph of
Solution:
x- intercepts:
y –intercepts:
The minimum value of y is – 4.
Problem solving
The graph of a quadratic function has the shape of a
parabola. Parabolas have many applications in engineering
and science. For example, radar, antennas and the paths of
projectiles are parabolic.
Example 15:
The rate of a chemical reaction, v, involving the transformation
of one compound into another compound is given by
where x is the concentration of the new compound. What value
of x results in the maximum value of v, and what is the
maximum value?
Solution:
So, the maximum value of v is 9 and it occurs when x = 3.
Example 16:
A farmer wants to build a fence around a rectangular paddock
using the straight edge of a canal as one side. She has 400 m
of fencing and wants to enclose the maximum area.
a. If the width of the paddock is x m, express the:
(i) length, l, of the paddock in terms of x
(ii) area, A, of the paddock in terms of x
b. Express A in square completion form.
c. Sketch the graph of A showing all key features.
d. Determine the length and width of the paddock for its area
to be maximised.
Solution:
c. The graph is a parabola opening downwards with axis of
symmetry x = 100 and vertex at (100, 20000).
x-intercepts:
A-intercept:
Quadratic Inequalities
If the highest power of the unknown in an inequality is 2, then it
is known as a quadratic inequality.
To solve a quadratic inequality, let the quadratic expression be
equal to y , sketch a graph of y against x, and then from the
graph, read off the x-values __ satisfy the inequality.
Example 17:
Solve
Solution:
x-intercept:
y-intercept:
From the graph we
find that y is positive,
when x < 1 or x > 6
Simultaneous equations
The simultaneous solution of a linear equation and a
quadratic equation is given by the point or points of
intersection of the line and parabola representing the
equations.
To find the solution, eliminate y (or x) by substitution; and
solve the quadratic equation thus formed in x (or y). Then find
the corresponding value(s) of y (or x).
Example 19:
Solve the following simultaneous equation:
Solution:
Example 20:
Solve the following simultaneous equations
Solution:
Problem Solving II
• Some practical problems involving two unknowns are solved
by finding the simultaneous solution of a linear and a
quadratic equation. To solve such problems we proceed as
follows:
• Let the two unknowns be x and y.
• Use the information provided in the problem to write two
simultaneous equations and solve the equations for x and y.
• Check the reasonableness of the solution and then state the
answer in words.
Example 21:
If the length of a rectangular kitchen is half the square of
its width and its perimeter is 48 m, find the dimensions of the
kitchen.
Solution:
Sketching quadratic inequalities
The parabola divides the
Cartesian plane into two regions.
One region is above the parabola
and the other is below the
parabola.
Example 22:
Solution:
x-intercepts
y-intercepts
Test point
Example 23:
Solution:
x-intercepts:
y-intercept:
Test point:
Intersection of half planes
The region common to two or more inequalities is useful
in the study of linear programming; and it can be depicted on
the Cartesian plane as illustrated in the following example.
Example 24:
Solution:
x-intercepts:
x-intercept:
y-intercept:
y-intercept:
The points of intersection are obtained by solving the
simultaneous equations:
Problem solving unit
Problem 13.1 A Spider and a Well
A spider is at the bottom of a well. Each morning she
spins a web to ascend 2 m up the well's wall, and then during
the day she slips 1 m down the wall. If the well is 10 m deep,
how long will it take for the spider to get out of the well?
You must justify your answer by using diagram(s) and/or tables.
Problem 13.2 Lockers
• A school in a certain remote area has five lockers that are
numbered 1, 2, 3, 4 and 5. On the first day of the first term, the
first student on the roll runs through the corridor and opens
every locker. The second student shuts every second locker,
the third student changes every third locker (i.e. opens it if it is
shut, shuts it if it is open), the fourth student changes every
fourth locker, and the fifth student changes every fifth locker.
• a. Which of the lockers are left open?
• b. Extend the question stated in part (a) to:
(i) 10 lockers and 10 students
(ii) 20 lockers and 20 students
(iii) 30 lockers and 30 students
(iv) 50 lockers and 50 students
(v) 101 lockers and 101 students
(vi) 1000 lockers and 1000 students
• Is there a pattern between the number of students and the
number of open lockers?
Generalise the above result assuming that there are n students.
Problem 13.3 Finite Difference Table
• From the table we can infer that:
• For any quadratic function, the second difference in a finite difference table
is always constant.
• A finite difference table enables us to find a rule for the quadratic function.
Example
Find a rule for the function specified by the following table of
values:
Solution:
Note:
A finite difference table must include x = 0. So, the value of c can be determined.
Using a graphics calculator
• Clear any lists within the STAT menu (and if there are graphs
in memory, clear them using the Y = menu). Select EDIT from
the STAT menu and enter x values under L1 and y values under
L2. Both lists should have the same number of entries.
• Select Plot 1 on from the STAT PLOT menu and turn the other
plots off. Highlight the second plot type that gives a line
graph. Choose L1 for the Xlist and L2 for the Ylist (if they are
not already highlighted).
• Choose Zoom Stat from the ZOOM menu to obtain the graph
of the data, which is a parabola. Then choose Quad Reg in the
CALC section of the STAT menu. The screen display shows
that the relationfits the data.
Question:
• Find a rule for the quadratic function specified by each of the
following tables and check your answers with a graphics
calculator.
Projects
Project 13.1 Squares and Square Roots
Formulas containing squares and square roots often occur in
science. To perform numerical computations we need to find
rational numbers that approximate the value of the squares
and/or square roots.
Extension
3.a. Use a ruler and protractor to draw a right-angled triangle ABC having AB = 2
cm,
BC = 1.5 cm and angle ABC = 90º.
b. Use a ruler to find the length of the hypotenuse of the right-angled triangle.
c. Verify your answer by finding the length of the hypotenuse using Pythagoras'
Theorem.
Project 13.2 Minimum Cost
In the accompanying diagram, a hotel is situated 3 km from a
highway. The owner of the hotel is thinking of connecting to a
gas supply for heating and cooking purposes. At present the
nearest gas pipe stops 7 km down the highway.
The cost of extending the gas pipe through the bush is three
times as much as the cost of extending the pipe along the
highway. How should the gas pipe be placed to minimise the
cost of connecting it to the hotel?
To investigate the cost of laying the gas pipe to the hotel, you
may proceed as follows:
1. Draw a graphical model (as displayed below) from the information given
above.
2.
The gas pipe is laid from P to R and then from R to H. If QR = x km,
express the following in terms of x:
a. the length of PR
b. the length of HR
3.
If the cost of extending the gas pipe along the highway is $1000 per
kilometre, find:
a. the cost of extending the gas pipe along the highway
b. the cost of extending the gas pipe from the highway to the hotel.
4.
Let the total cost of extending the gas pipe from P to R and from R to H
be $C. Show that C is given by
Modelling with a TI-82/83 Graphics Calculator
A graphics calculator can be used to determine the minimum and/or
maximum values of a function. In this project, you are required to find the
minimum cost of extending the gas pipe to the hotel. You may proceed as
follows:
To create a table of values of Y1 (i.e. C) for different values of x, press Table.
Using this table determine:
a. the value of x which gives the minimum value of C
b. the minimum cost of extending the gas pipe.
6. Use trigonometry to find the angle QRH.
7. Display the measurements for the lengths of PR, QR, HR and the angle
QRH in the following diagram.
Modelling with Excel
Microsoft® Excel spreadsheets can be used quite effectively to generate a
table of values. In this project, you are required to find the minimum cost
of extending the gas pipe. You may proceed as follows:
Open a new spreadsheet, and type x in the cell A1 and C in the cell B1.
Type 0 in the cell A2. Then type the formula = A2 + 0.05 into the cell A3 and
fill down (using the edit fill down command) this formula to an appropriate
row.
Type the formula for the cost C in the cell B2 and fill down the formula to an
appropriate row.
8. Using the values generated in the spreadsheet determine:
a. the value of x which gives the minimum value of C
b. the minimum cost of extending the gas pipe.
9. Use trigonometry to find the angle QRH.
10. Display the measurements for the lengths of PR, QR, RH and the angle
QRH in the following diagram.
Write a brief report on your findings by comparing the results obtained from
the two different methods.
Project 13.3 Area under the Curve
We know that:
In this project, you are required to determine the area that is
above the x-axis and under the parabola
We do not know a formula that can be used to find the
shaded area. However, we can find an approximate value of
the shaded area enclosed under the curve and above the x-axis
by dividing it into small regions, say rectangles, as shown
below.
It is clear from the diagram that the area under the curve and
above the x-axis is approximately equal to the sum of the two
rectangles. That is:
So, the shaded area is approximately 50 square units.
In the above example, the width of each rectangle is 5 units. This
is known as the strip width.
We can obtain a better approximation to the area under the curve
of a function by increasing the number of rectangles (i.e.
strips). Reducing the width of the rectangles (i.e. strips)
increases the number of rectangles.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 2 units. That is, divide the shaded area into 5 rectangles.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 1 unit. That is, divide the shaded area into 10 rectangles.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 0.5 units.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 0.4 units.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 0.2 units.
Determine the area under the curve
and above the x-axis by dividing the area into rectangles of
width 0.1.
Congratulations
now run away