Functions and Their

Graphs

Types of Functions

2. Quadratic FunctionsA quadratic function is a function of the form f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0.Domain: the set of real numbersGraph: parabolaExamples: parabolas parabolas

opening upward opening downward

The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph below. If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward. The vertex (or turning point) is the minimum or maximum point.

Graphs of Quadratic Functions

Graphing Parabolas Given f(x) = ax2 + bx +c1. Determine whether the parabola opens upward or

downward. If a > 0, it opens upward. If a < 0, it opens downward.

2. Determine the vertex of the parabola. The vertex is

3. The axis of symmetry is

The axis of symmetry divides the parabola into two equal parts such that one part is a mirror image of the other.

a

bacab

44,

2

2

abx2

Graphing Parabolas Given f(x) = ax2 + bx +c4. Find any x-intercepts by replacing f (x) with 0.

Solve the resulting quadratic equation for x. 5. Find the y-intercept by replacing x with zero. 6. Plot the intercepts and vertex. Connect these

points with a smooth curve that is shaped like a cup.

Graph: f(x) = x2 x 2 1. Determine the values of a, b, c. a = 1, b = -1, c = -2 2. Determine whether the parabola opens

upward or downward. Since a > 0, then parabola opens

upward. 3. Find the vertex of the parabola.

abx2

21

)1(2)1(

x

492

21

21

21 2

f

Graph: f(x) = x2 x 2 4. Find the intercepts, if any. x – intercepts: If y = 0, then x = 2, – 1. y – intercept: If x = 0, then y = –2. 5. Plot the intercepts and vertex. Connect

these points with a smooth curve that is shaped like a cup.

Graph: f(x) = x2 x 23

2

1

-1

-2

-3

-6 -4 -2 2 4 6

(0.5,-2.25)

(2,0)(-1,0)

(0,-2)

Graph: f(x)=15 2xx2 1. Determine the values of a, b, c. a = -1, b = -2, c = 15 2. Determine whether the parabola opens

upward or downward. Since a < 0, then parabola opens

downward. 3. Find the vertex of the parabola. vertex (-1, 16)

Graph: f(x)=15 2xx2 4. Find the intercepts, if any. x – intercepts: If y = 0, then x = 3, –5. y – intercept: If x = 0, then y = 15. 5. Plot the intercepts and vertex. Connect

these points with a smooth curve that is shaped like a cup.

Graph: f(x)=15 2xx2 16

14

12

10

8

6

4

2

-15 -10 -5 5 10 15(3,0)(-5,0)

(-1,16)

Example The function f(x) = 1 - 4x - x2 has its

vertex at _____.  A. (2,11) B. (2,-11) C.( -2,-3) D.(-2,5)

Example Identify the graph of the given function: y = 3x2 -

3.

Example Identify the graph of the given function: 4y = x2.

Example Identify the graph of the given function: y = (x -

2)(x – 2).

Graph of : f(x)=ax2 + bx +c, a ≠ 0

Parabola

a > 0 opens upward

a < 0 opens downward

Vertex (-b/2a, f(-b/2a)

b2-4ac > 0 2 x – intercepts

b2-4ac = 0 1 x – intercept

b2-4ac < 0 No x – intercept

X – intercepts (x, 0)

Y – intercept c

Standard Form of Quadratic Functions

A quadratic function of the form f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0, is in standard form if it is written as f(x) = a(x – h)2 + k , a ≠ 0.

The vertex is at (h, k).

Example: Express f(x) = x2 - x - 2 in standard form. Solution: f(x) = (x2 - x) – 2 By completing the square, f(x) = (x2 - x + (-1/2)2) – 2 - (-1/2)2

f(x) = (x - 1/2)2 – 2 - (-1/2)2

f(x) = (x – 0.5)2 – 2.25 Where (h, k) = (0.5, -2.25)

Graph: f(x) = x2 x 23

2

1

-1

-2

-3

-6 -4 -2 2 4 6

(0.5,-2.25)

(2,0)(-1,0)

(0,-2)

Example: Express f(x) = 15 – 2x – x2 in standard

form. Solution: f(x) = – (x2 + 2x) + 15 By completing the square, f(x) = –(x2 +2x + (1)2) + (15 +1) f(x) = –(x + 1)2 + 16 Where (h, k) = (-1, 16)

Graph: f(x)=15 2xx2 16

14

12

10

8

6

4

2

-15 -10 -5 5 10 15(3,0)(-5,0)

(-1,16)

Exercises: (page 37) Graph each of the given equation. Find its vertex,

axis of symmetry, x and y intercepts, and domain and range.

2. f(x) = x2 + x – 2 3. f(x) = – x2 + 10x – 25 4. f(x) = – 4x2 - 20x – 24 5. f(x) = 6x2 – 7x – 5

More on Parabolas

Relations and Functions

More on Parabolas Parabola Opens Upward/Downward

o f(x) = a(x – h)2 + k , a ≠ 0o (y – k) = a(x – h)2 , a ≠ 0

Parabola Opens to the right/ left x = a(y – k)2 + h , a ≠ 0 (x – h) = a(y – k)2 , a ≠ 0

Graph the following: 1. x2 = 16y 2. y2 = – 12x 3. 2x2 + 5y = 0 4. (x – 3 )2 = 10 (y + 2) 5. y2 – 12x +48 = 0

Graph the following: 1. (x + 3 )2 – 8 (y + 6) = 02. y2 = – 32x3. x2 – 2x – y – 1 = 04. y2 – 4x – 8y + 24 = 05. y2 – 4x – 8y + 7 = 06. x2 – 4x – 12y – 32 = 07. x2 + 4x – 16y +4 = 0

Exercises

More Exercises: Activity Sheet 1.4 pages 365 – 366 # 1, 2,

3, 4, 5

References: (Online Graphing Utility)

http://rechneronline.de/function-graphs/ http://www.coolmath.com/graphit/