**Quadratic functions** and the properties of their graphs such as vertex and

You can also use this applet to explore the relationship between the **quadratic function**

Once you finish the present tutorial, you may want to go through tutorials on **quadratic functions** , graphing **quadratic functions** and Solver to Analyze and Graph a Quadratic Function

If needed, Free graph paper is available.

A

where

__Examples of quadratic functions__

f(x) = -2x ^{ 2}+ x - 1f(x) = x ^{ 2}+ 3x + 2

__Interactive Tutorial (1)__

Explore quadratic functions interactively using an html5 applet shown below; press "draw' button to start

- Use the boxes on the left panel of the applet window to set coefficients
a ,b andc to the values in the examples above, 'draw' and observe the graph obtained. Note that the graph corresponding to part a) is a parabola opening down since coefficient a is negative and the graph corresponding to part b) is a parabola opening up since coefficient a is positive. You may change the values of coefficient a, b and c and observe the graphs obtained.

Any

where

Let us start with the

- Given function
f(x)

f(x) = ax ^{ 2}+ bx + c

- factor coefficient
a out of the terms inx and^{ 2}x

f(x) = a ( x ^{ 2}+ (b / a) x ) + c

- add and subtract
(b / 2a) inside the parentheses^{ 2}

f(x) = a ( x ^{ 2}+ (b/a) x + (b/2a)^{ 2}- (b/2a)^{ 2}) + c

- Note that

x ^{ 2}+ (b/a) x + (b/2a)^{ 2}

- can be written as

(x + (b/2a)) ^{ 2}

- We now write
f as follows

f(x) = a ( x + (b / 2a) ) ^{ 2}- a(b / 2a)^{ 2}+ c

- which can be written as

f(x) = a ( x + (b / 2a) ) ^{ 2}- (b^{ 2}/ 4a) + c

- This is the standard form of a
**quadratic function**with

h = - b / 2a

k = c - b ^{ 2}/ 4a

- given function

f(x) = -2 x ^{ 2}+ 4x + 1

- factor -2 out

f(x) = -2(x ^{ 2}- 2 x) + 1

- We now divide the coefficient of
x which is-2 by 2 and that gives-1 .

f(x) = -2(x ^{ 2}- 2x + (-1)^{ 2}- (-1)^{ 2}) + 1

- add and subtract
(-1) within the parentheses^{ 2}

f(x) = -2(x ^{ 2}- 2x + (-1)^{ 2}) + 2 + 1

- group like terms and write in standard form

f(x) = -2(x - 1) ^{ 2}+ 3

- The above gives
h = 1 andk = 3 .

h andk can**also**be found using the formulas forh andk obtained above.

h = - b / 2a = - 4 / (2(-2)) = 1

k = c - b ^{ 2}/ 4a = 1 - 4^{ 2}/(4(-2))= 3

- The vertex of the graph is at
(1,3) .

- Go back to the applet window and set a to
-2 ,b to4 andc to1 (values used in the above example). Check that the graph opens down (a < 0 ) and that the vertex is at the point(1,3) and is a maximum point. - Use the applet window and set a to
1 ,b to-2 andc to0 ,f(x) = x . Check that the graph opens up (^{ 2}- 2 xa > 0 ) and that the vertex is at the point(1,-1) and is a minimum point.

The

are the

The above equation has two real solutions and therefore the graph has

and

f(x) = x _{ 2}+ 2 x - 3g(x) = -x _{ 2}+ 2 x - 1h(x) = -2 _{ 2}+ 2 x - 2

- To find the
x intercepts, we solve

x ^{ 2}+ 2 x - 3 = 0

discriminantD = 2 ^{ 2}- 4 (1)(-3) = 16

two real solutions:

x _{1}= (-2 + √16) / (2 * 1) = 1

and

x _{2}= (-2 - √16) / (2 * 1) = -3

The graph of function in part a) has twox intercepts are at the points(1,0) and(-3,0) .

- We solve
-x ^{ 2}+ 2 x - 1 = 0

discriminantD = 2 ^{ 2}- 4(-1)(-1) = 0

one repeated real solutionsx_1 = -b / 2a = -2 / -2 = 1

The graph of function in part b) has onex intercept at(1,0) .

- We solve
-2 x ^{ 2}+ 2 x - 2 = 0

discriminantD = 2 ^{ 2}- 4(-2)(-2) = -12

No real solutions for the above equation

No x intercept for the graph of function in part c).

- Go to the applet window and set the values of
a ,b andc for each of the examples in partsa ,b andc above and check the discriminant and thex intercepts of the corresponding graphs. - Use the applet window to find any
x intercepts for the following quadratic functions.

a)f(x) = x ^{ 2}+ x - 2

b)g(x) = 4 x ^{ 2}+ x + 1

a)h(x) = x ^{ 2}- 4 x + 4

Use the analytical method described in the above example to find the x intercepts and compare the results. - Use the applet window and set
a ,b andc to values such thatb . How many^{ 2}- 4 a c < 0x -intercepts does the graph off(x) have ? - Use the applet window and set
a ,b andc to values such thatb . How many^{ 2}- 4 a c = 0x -intercepts the does the graph off(x) have? - Use the applet window and set
a ,b andc to values such thatb . How many^{ 2}- 4ac > 0x -intercepts does the graph off(x) have ?

The y intercept of the graph of a quadratic function is given by

f(x) = x ^{ 2}+ 2 x - 3g(x) = 4 x ^{ 2}- x + 1h(x) = -x ^{ 2}+ 4 x + 4

f(0) = -3 . The graph off has ay intercept at(0,-3) .

g(0) = 1 . The graph ofg has ay intercept at(0,1) .

h(0) = 4 . The graph ofh has ay intercept at(0,4) .

- Use the applet window to check the
y intercept for the**quadratic functions**in the above example. - Use the applet window to check the
y intercept is at the point(0,c) for different values ofc .

As an exercise you are asked to find the equation of a quadratic function whose graph is shown in the applet and write it in the form

USE this applet to Find Quadratic Function Given its Graph

Example: Find the graph of the quadratic function f whose graph is shown below.

Solution

There are several methods to answer the above question but all of them have one idea in common: you need to understand and then select the right information from the graph.

The above graph has two

We now use the

and solve for

The above parabola has a vertex at

We use the

Since a quadratic function has the form

we need 3 points on the graph of

The following points are on the graph of

point

solve for

The two other points give two more equations

which leads to

and

which becomes

Solve the last two equations in a and b to obtain

Go back to the applet above, generate a graph and find its equation. You can generate as many graphs, and therefore questions, as you wish.

More on **quadratic functions** and related topics

- Derivatives of Quadratic Functions: Explore the quadratic function
*f(x) = ax*and its derivative graphically and analytically.^{ 2}+ b x + c - Match Quadratic Functions to Graphs. Excellent activity where quadratic functions are matched to graphs.
- Find Vertex and Intercepts of
**Quadratic Functions**- Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function. - Tutorial on
**Quadratic Functions**(1).

**Quadratic Functions**- Problems (1).

- graphing
**quadratic functions**.

**quadratic functions**in vertex form .

- Quadratic Functions Transformations