and the properties of their graphs such as **vertex** and x and y **intercepts** are explored, interactively, using an applet.

The graph of a **quadratic function** is "U" shaped and is called a **parabola**. The exploration is carried by changing the values of all 3 coefficients $a$, $h$ and $k$. Once you finish the present tutorial, you may want to go through another tutorial on graphing quadratic functions.

This form of the quadratic function is also called the vertex form.

## A - Vertex, maximum and minimum values of a quadratic function

^{2}+ k

The term (x - h)

^{2}is a square, hence is either positive or equal to zero.

(x - h)

^{2}≥ 0

If you multiply both sides of the above inequality by coefficient

**a**, there are two possibilitiesto consider, a is positive or a is negative.

case 1: a is positive

a(x - h)

^{2}≥ 0.

Add k to the left and right sides of the inequality

a(x - h)

^{2}+ k ≥ k.

The left side represents f(x), hence f(x) ≥ k. This means that k is the

**minimum**value of function f.

case 2: a is negative

a(x - h)

^{2}≤ 0.

Add k to the left and right sides of the inequality

a(x - h)

^{2}+ k ≤ k.

The left side represents f(x), hence f(x) ≤ k. This means that k is the

**maximum**value of function f.

Note also that k = f(h), hence point (h,k) represents a

**minimum**point when a is positive and a

**maximum**point when a is negative. This point is called the

**vertex**of the graph of f.

__Example:__

Find the **vertex** of the graph of each function and identify it as a **minimum** or **maximum** point.

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

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

c) f(x) = 2(x - 3)^{2}

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

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

^{2}- 1

a = -1 , h = -2 and k = -1. The

**vertex**is at (-2,-1) and it is a

**maximum**point since a is negative.

b) f(x) = -x

^{2}+ 2 = -(x - 0)

^{2}+ 2

a = -1 , h = 0 and k = 2.

**The vertex**is at (0,2) and it is a

**maximum**point since a is negative.

c) f(x) = 2(x - 3)

^{2}= 2(x - 3))

^{2}+ 0

a = 2 , h = 3 and k = 0. The

**vertex**is at (3,0) and it is a minimum point since a is positive.

__Interactive Tutorial__

Use the html 5 (better viewed using chrome, firefox, IE 9 or above) applet below to explore the graph of a quadratic function in vertex form: f(x)=a (x-h)

^{2}+ k where the coefficients a, h and k may be changed in the applet below. Enter values in the boxes for a, h and k and press draw.

1 - Use the boxes on the left panel of the applet to set a to -1, h to -2 and k to 1. Check theposition of the

**vertex**and whether it is a

**minimum**or a

**maximum**point. Compare to part a) in the example above.

2 - Set a to -1, h to 0 and k to 2. Check the position of the

**vertex**andwhether it is a

**minimum**or a

**maximum**point. Compare to part b) in the example above.

3 - Set a to 2, h to 3 and k to 0. Check the position of the

**vertex**and whether it is a

**minimum**or a

**maximum**point. Compare to part c) in the example above.

4 - Set h and k to some values and a to positive values only. Check that the

**vertex**is always a

**minimum**point.

5 - Set h and k to some values and a to negative values only. Check that the

**vertex**is always a

**maximum**point.

## B - x intercepts of the graph of a quadratic function in standard form

The x **intercepts** of the graph of a **quadratic function** f given by

^{2}+ k

are the

__real__

solutions, if they exist, of the **quadratic equation**

^{2}+ k = 0

add -k to both sides

a(x - h)

^{2}= -k divide both sides by a

(x - h)

^{2}= -k / a

The above equation has real solutions if - k / a is positive or zero.

The solutions are given by

x

_{1}= h + √(- k / a)

x

_{2}= h - √(- k / a)

__Example:__

Find the x **intercepts** for the graph of each function given below

a) f(x) = -2(x - 3)

^{2}+ 2

b) g(x) = -(x + 2)

^{2}

c) h(x) = 4(x - 1)

^{2}+ 5

a) To find the x

**intercepts**, we solve

-2(x - 3)

^{2}+ 2 = 0

-2(x - 3)

^{2}= -2

(x - 3)

^{2}= 1

two real solutions: x

_{<1}= 3 + √1 = 4 and x

_{2}= 3 - √1 = 2

The graph of function in part a) has two x

**intercepts**are at the points (4,0) and (2,0)

b) We solve

-(x + 2)

^{2}= 0

one repeated real solution x

_{1}= - 2

The graph of function in part b) has one $x$

**intercept**at (-2,0).

c) We solve

4(x - 1)

^{2}+ 5 = 0

- k / a = - 5 / 4 is negative. The above equation has no real solutions and the graph of function h hasno x intercept.

__Interactive Tutorial__

1 - Go back to the applet window and set the values of a, h and k for each of the examples inparts a, b and c above and check the the x

**intercepts**of the corresponding graphs.

2 - Use the applet window to find any x

**intercepts**for the following functions. Use theanalytical method described in the above example to find the x

**intercepts**and compare theresults.

a) f(x) = 5(x - 3)

^{2}+ 3

b) g(x) = -(x + 2)

^{2}+ 1

c) h(x) = 3(x - 1)

^{2}

3 - Use the applet window and set a and k to values such that -k / a < 0.How many

**x-intercepts**the graph of f has ?

4 - Use the applet window and set k to zero.How many

**x-intercepts**the graph of f has ?

5 - Use the applet window and set a and k to values such that - k / a > 0.How many

**x-intercepts**the graph of f has ?

## C - From vertex form to general form with a, b and c.

Rewriting the vertex form of a quadratic function into the general form is carried out by expanding the square in the vertex form and grouping like terms.

__Example:__

Rewrite f(x) = -(x - 2)^{2} - 4 into general form with coefficients a, b and c.

Expand the square in f(x) and group like terms

f(x) = -(x - 2)

^{2}- 4 = -(x

^{2}-4 x + 4) - 4

= - x

^{2}+ 4 x - 8

A tutorial on how to

find the equation of a quadratic function given its graph can be found in this site.More references on quadratic functions and their properties.

- 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**. - Match Quadratic Functions to Graphs. Excellent activity where quadratic functions are matched to graphs.
- graphing quadratic functions .
- quadratic functions (general form) .