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Graphing Quadratic Functions: Parabolas MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Graphing Quadratic Functions: Parabolasbanach.millersville.edu/~bob/math101/Parabolas/main.pdf · Remarks For all quadratic functions, the domain is the set of all real numbers. For

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Graphing Quadratic Functions: ParabolasMATH 101 College Algebra

J. Robert Buchanan

Department of Mathematics

Summer 2012

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Objectives

In this lesson we will learn to:

graph a parabola (a quadratic function) and determine itsvertex, domain, range, line of symmetry, and zeros, and

solve applied problems by using quadratic functions andthe concept of maximum and minimum.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Quadratic Functions

Definition

Any function that can be written in the form

y = ax2 + bx + c

where a, b, and c are real numbers with a 6= 0 is a quadraticfunction.

Remarks:

The graph of any quadratic function is a curve called aparabola.

Vertical parabolas are parabolas which “open upward” or“open downward”.

Horizontal parabolas are parabolas which “open left” or“open right”.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Example

Consider the function y = x2 − 6x + 5. We can choose somevalues for x and compute the corresponding values for y .

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Example

Consider the function y = x2 − 6x + 5. We can choose somevalues for x and compute the corresponding values for y .

x y−1 12

0 51 02 −33 −44 −35 06 57 12

-2 2 4 6 8x

5

10

15

20

y

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Anatomy of a Parabola

Vertex: H3,-4L

x=3

-2 2 4 6 8x

-5

5

10

15

20

y

Observations:

The turning point at (3,−4) is called the vertex of theparabola.

The vertical line at x = 3 is called the line of symmetry ofthe parabola.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Functions of the Form y = ax2

Consider the graphs of the following parabolas of the formy = ax2.

-3

-1

-

1

3

1

2

1

2

-4 -2 2 4x

-40

-20

20

40

y

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Remarks

For all quadratic functions, the domain is the set of all realnumbers.For parabolas of the form y = ax2,

the parabola opens upward if a > 0 and the range is y ≥ 0,the parabola opens downward if a < 0 and the range isy ≤ 0,the vertex is the point (0, 0),the line of symmetry is x = 0,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Functions of the Form y = ax2 + k

Consider the graphs of the following parabolas of the formy = ax2 + k (for these examples a = ±1).

-2

0

1

-2 -1 1 2x

-2

-1

1

2

3

4y

-2

01

-2 -1 1 2x

-4

-3

-2

-1

1

2y

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Remarks

Adding k to ax2 shifts or translates the graph verticallyby k units.For parabolas of the form y = ax2 + k ,

the parabola opens upward if a > 0 and the range is y ≥ k ,the parabola opens downward if a < 0 and the range isy ≤ k ,the vertex is the point (0, k),the line of symmetry is x = 0,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Functions of the Form y = a(x − h)2

Consider the graphs of the following parabolas of the formy = a(x − h)2 (for these examples a = ±1).

-10 -5 5 10x

50

100

150

200

y

4

0

-5

-10 -5 5 10x

-200

-150

-100

-50

y

-4

05

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Remarks

Subtracting h from x in a(x − h)2 shifts or translates thegraph horizontally by h units.

For parabolas of the form y = a(x − h)2,the parabola opens upward if a > 0 and the range is y ≥ 0,the parabola opens downward if a < 0 and the range isy ≤ 0,the vertex is the point (h, 0),the line of symmetry is x = h,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Functions of the Form y = ax2 + bx + c

y = ax2 + bx + c

= a(

x2 +ba

x)

+ c

= a(

x2 +ba

x +b2

4a2 −b2

4a2

)

+ c

= a(

x +b2a

)2

−b2

4a+ c

= a(x − h)2 + k

where

h = −b2a

k =4ac − b2

4a

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Functions of the Form y = ax2 + bx + c

y = ax2 + bx + c

= a(

x2 +ba

x)

+ c

= a(

x2 +ba

x +b2

4a2 −b2

4a2

)

+ c

= a(

x +b2a

)2

−b2

4a+ c

= a(x − h)2 + k

where

h = −b2a

k =4ac − b2

4aRemark: we can obtain the graph of y = ax2 + bx + c througha horizontal shift of h units and a vertical shift of k units.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Examples

The following graphs show parabolas of y = a(x − h)2 + k fordifferent choices of h and k . The values of a used are ±1.

-4 -2 2 4x

20

40

60

80

y

H0,0L

H-3,0L

H-3,20L

H0,20L

-6 -4 -2 2 4 6x

-100

-80

-60

-40

-20

y

H0,-10L

H0,15L

H-4,-10LH-4,15L

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Remarks

Writing the standard form of a quadratic function

y = ax2 + bx + c = a(x − h)2 + k

where h = −b2a

and k =4ac − b2

4a

Subtracting h from x in a(x − h)2 shifts or translates thegraph horizontally by h units.Adding k to a(x − h)2 shifts or translates the graphvertically by k units.The parabola opens upward if a > 0 and the range isy ≥ k ,The parabola opens downward if a < 0 and the range isy ≤ k ,The vertex is the point (h, k), andThe line of symmetry is x = h.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Zeros of a Quadratic Function

If the graph of the quadratic function y = ax2 + bx + c crossesthe x-axis, these points are called the zeros of the function.They can be found by solving the equation:

ax2 + bx + c = 0.

Example

Find the zeros of the function, the line of symmetry, thecoordinates of the vertex, the range, and the graph of theparabola for

y = x2 − 7x + 10

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Solution

Zeros of the function:

0 = x2 − 7x + 10 = (x − 2)(x − 5)

x = 2 or x = 5

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Solution

Zeros of the function:

0 = x2 − 7x + 10 = (x − 2)(x − 5)

x = 2 or x = 5

y = x2 − 7x + 10 = x2 − 7x +494

−494

+ 10

=

(

x −72

)2

−494

+ 10 =

(

x −72

)2

−94

Line of symmetry: x =72

Vertex:(

72

,−94

)

Domain: all real numbers (−∞,∞)

Range: y ≥ −94

since a = 1 > 0.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Graph

Vertex: H7�2,-9�4L

x=7�2

2 4 6 8x

-5

5

10

15

20

y

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Maximum and Minimum Values

Observations: for parabolas which open upward, the vertex isthe lowest point on the parabola, while for parabolas whichopen downward, the vertex is the highest point on the parabola.

J. Robert Buchanan Graphing Quadratic Functions: Parabolas

Maximum and Minimum Values

Observations: for parabolas which open upward, the vertex isthe lowest point on the parabola, while for parabolas whichopen downward, the vertex is the highest point on the parabola.

Theorem

For a parabola with its equation in the form y = a(x − h)2 + k,1 if a > 0, the parabola opens upward and (h, k) is the

lowest point and y = k is called the minimum value of thefunction,

2 if a < 0, the parabola opens downward and (h, k) is thehighest point and y = k is called the maximum value ofthe function,

J. Robert Buchanan Graphing Quadratic Functions: Parabolas