Quadratic Functions(General Form)

Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an html5 applet.

You can also use this applet to explore the relationship between the x intercepts of the graph of a quadratic function f(x) and the solutions of the corresponding quadratic equation f(x) = 0. The exploration is carried by changing values of 3 coefficients a, b and c included in the definition of f(x).

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 - Definition of a quadratic function


A
quadratic functionf is a function of the form
f(x) = ax 2 + bx + c

where
a , b and c are real numbers and a not equal to zero. The graph of the quadratic function is called a parabola. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a .

Examples of quadratic functions

  1. f(x) = -2x 2 + x - 1
  2. f(x) = x 2 + 3x + 2

Interactive Tutorial (1)

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

a =
-10+10

b =
-10+10

c =
-10+10

>

  • Use the boxes on the left panel of the applet window to set coefficients a, b and c 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.

    Answers


B - Standard form of a quadratic function and vertex


Any
quadratic function can be written in the standard form

f(x) = a(x - h) 2 + k


where
h and k are given in terms of coefficients a , b and c .

Let us start with the
quadratic function in general form and complete the square to rewrite it in standard form.
  • Given function f(x)
    f(x) = ax 2 + bx + c

  • factor coefficient a out of the terms in x 2 and x

    f(x) = a ( x 2 + (b / a) x ) + c

  • add and subtract (b / 2a) 2 inside the parentheses
    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


When you graph a quadratic function , the graph will either have a maximum or a minimum point called the vertex. The x and y coordinates of the vertex are given by h and k respectively.

Example : Write the quadratic functionf given by f(x) = -2 x 2 + 4 x + 1 in standard form and find the vertex of the graph.

Solution
  • 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) 2 within the parentheses
    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 and k = 3.

  • h and k can also be found using the formulas for h and k 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).

Interactive Tutorial (2)

  • Go back to the applet window and set a to -2, b to 4 and c to 1 (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 and c to 0, f(x) = x 2 - 2 x. Check that the graph opens up (a > 0) and that the vertex is at the point (1,-1) and is a minimum point.


C - x intercepts of the graph of a quadratic function



The
x intercepts of the graph of a quadratic function f given by
f(x) = a x 2 + b x + c

are the
real solutions, if they exist, of the quadratic equation
a x 2 + b x + c = 0


The above equation has two real solutions and therefore the graph has
x intercepts when the discriminant D = b^2 - 4 a c is positive. It has one repeated solution when D is equal to zero. The solutions are given by the quadratic formulas

x 1 = (-b + √ D)/(2 a)

and
x 2 = (-b - √ D)/(2 a)


Example: Find the x intercepts for the graph of each function given below

  1. f(x) = x 2 + 2 x - 3
  2. g(x) = -x 2 + 2 x - 1
  3. h(x) = -2 2 + 2 x - 2

Solution
  1. To find the x intercepts, we solve

    x 2 + 2 x - 3 = 0

    discriminant D = 2 2 - 4 (1)(-3) = 16

    two real solutions:
    x1 = (-2 + √16) / (2 * 1) = 1
    and
    x2 = (-2 - √16) / (2 * 1) = -3

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

  2. We solve -x 2 + 2 x - 1 = 0

    discriminant D = 2 2 - 4(-1)(-1) = 0

    one repeated real solutions x_1 = -b / 2a = -2 / -2 = 1

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

  3. We solve -2 x 2 + 2 x - 2 = 0

    discriminant D = 2 2 - 4(-2)(-2) = -12

    No real solutions for the above equation

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

Interactive Tutorial (3)

  • Go to the applet window and set the values of a, b and c for each of the examples in parts a, b and c above and check the discriminant and the x 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 and c to values such that b 2 - 4 a c < 0. How many x-intercepts does the graph of f(x) have ?
  • Use the applet window and set a, b and c to values such that b 2 - 4 a c = 0. How many x-intercepts the does the graph of f(x) have?
  • Use the applet window and set a, b and c to values such that b 2 - 4ac > 0. How many x-intercepts does the graph of f(x) have ?

Answers


D - y intercepts of the graph of a quadratic function



The y intercept of the graph of a quadratic function is given by
f(0) = c .

Example: Find the y intercept of the graph of the following quadratic functions.
  1. f(x) = x 2 + 2 x - 3
  2. g(x) = 4 x 2 - x + 1
  3. h(x) = -x 2 + 4 x + 4
Solution
  1. f(0) = -3. The graph of f has a y intercept at (0,-3).
  2. g(0) = 1. The graph of g has a y intercept at (0,1).
  3. h(0) = 4. The graph of h has a y intercept at (0,4).
Interactive Tutorial (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 of c.

E - Exercises: Find the equation of a quadratic function given its graph



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 f(x) = a x 2 + b x + c.

USE this applet to Find Quadratic Function Given its Graph

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

graphical solution to check


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.

method 1:

The above graph has two
x intercepts at (-3,0) and (-1,0) and a y intercept at (0,6) . The x coordinates of the x intercepts can be used to write the equation of function f as follows:

f(x) = a(x + 3)(x + 1)

We now use the
y intercept f(0) = 6

6 = a(0 + 3)(0 + 1)

and solve for
a to find a = 2 . The formula for the quadratic function f is given by :

f(x) = 2(x + 3)(x + 1) = 2 x 2 + 8 x + 6

method 2:

The above parabola has a vertex at
(-2 , -2) and a y intercept at (0,6) . The standard (or vertex) form of a quadratic function f can be written

f(x) = a(x + 2) 2 - 2

We use the
y intercept f(0) = 6

6 = a(0 + 2) 2 - 2 . Solve for a to find a = 2 . The formula for the quadratic function f is given by :

f(x) = 2(x + 2) 2 - 2 = 2 x 2 + 8 x + 6

method 3:

Since a quadratic function has the form

f(x) = a x 2 + b x + c

we need 3 points on the graph of
f in order to write 3 equations and solve for a , b and c .

The following points are on the graph of
f

(-3 , 0) , (-1 , 0) and (0 , 6)

point
(0 , 6) gives

f(0) = 6 = a * 0 2 + b * 0 + c = c
solve for
c to obtain c = 6
The two other points give two more equations

(-3 , 0) gives f(-3) = a * (-3) 2 + b * (-3) + 6

which leads to
9 a - 3 b + 6 = 0

and
(-1 , 0) gives f(-3) = a (-1) 2 + b * (-1) + 6

which becomes
a - b + 6 = 0

Solve the last two equations in a and b to obtain

a = 2 and b = 8 and gives

f(x) = 2 x 2 + 8 x + 6

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.

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