Quadratic FunctionsFUNDAMENTAL COACHING CENTRE( FCC)

Quadratic functionsA 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 1 3y 9 1 0 1 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 left-hand 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 -2 -1 0 1 2y 2 0.5 0 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 offor -3 ≤ x ≤ 3

x -3 -1 0 1 3y -9 -1 0 -1 -9

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).

Example 4: Plot the graph of for -2 ≤ x ≤ 2

x -2 -1 0 1 2y -8 -2 0 -2 -8

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).

Graphs of Example 5:Plot the graph of for -2 ≤ x ≤ 2

x -2 -1 0 1 2y 7 4 3 4 7

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).

Graphs of Example 6:Plot the graph offor -3 ≤ x ≤ 3

x -3 -1 0 1 3y -8 0 1 0 -8

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).

Graphs of Example 7:Plot the graph offor 0 ≤ x ≤ 4

x 0 1 2 3 4y 4 1 0 1 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).

Graphs of Example 8:Plot the graph offor 0 ≤ x ≤ 4

x 0 1 2 3 4y -4 -1 0 -1 -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).

Graphs of Example 9:Plot the graph offor -1 ≤ x ≤ -3

x -1 0 1 2 3y 6 3 2 3 6

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).

Graphs ofExample 10:Plot the graph of for -1 ≤ x ≤ -3

x -1 0 1 2 3y -2 1 2 1 -2

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).

Quadratic graphs by transformationsIt is useful to consider some graphs as the dilation,

translation or reflection of a basic graph.

Dilation and ReflectionDilation has the effect of stretching or compressing a graph.  Dilation is either from the x-axis or they-axis.  Also, it could be from both axes.

Consider the graphs of the following functions on the same set of axes:

We notice that:

TranslationA 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)²Consider 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 solvingThe 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 xb.  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 InequalitiesIf 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 equationsThe 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:

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 planesThe 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-intercept:

y-intercept:

x-intercepts:

y-intercept:

The points of intersection are obtained by solving the simultaneous equations:

Problem solving unitProblem 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.

ExampleFind 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.

ProjectsProject 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.

Extension3.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 CostIn 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 PRb.  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 highwayb.  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 Cb.  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 Cb.  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 CurveWe 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.

Congratulationsnow run away