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### Graphing Quadratic Functions

- A
*quadratic function*is a function of the form f(x) = ax^{2}+ bx + c, where a ≠ 0. This is called the*standard form*of a quadratic function. - The graph of a quadratic function is a
*parabola*. If a > 0, the parabola opens upward. If a < 0, it opens downward. - The
*axis of symmetry*divides a parabola into two symmetrical halves. Its equation is x = -b/2a. - Use the axis of symmetry to help you graph a parabola. Graph the right or left half and then reflect the graph across the axis of symmetry.
- The maximum or minimum value of the graph occurs at the
*vertex*. The formula for the x-coordinate of the vertex is x = -b/2a. Use this formula to find the maximum or minimum.

### Graphing Quadratic Functions

Find the maximum/minimum point of the function y = - x

^{2}+ 4x + 5- Find the vertex using x = − [b/2a]x = − ( [4/(2( − 1))] ) = 2y = − ( 2 )
^{2}+ 4( 2 ) + 5 = 9 - Determine which direction the parabola opens

a < 0 opens downwards - Consider vertex and direction to determine maximum/minimum

(2,9) is a maximum

Find the maximum/minimum point of the function y = x

^{2}− 2x − 6- Find the vertex using x = − [b/2a]x = − ( [( − 2)/2(1)] ) = 1y = ( 1 )
^{2}− 2( 1 ) − 6 = − 7 - Determine which direction the parabola opens

a > 0 opens upwards - Consider vertex and direction to determine maximum/minimum

(1, - 7) is a minimum

Find the maximum/minimum point of the function y = [(x

^{2})/4] + x- Find the vertex using x = − [b/2a]x = − ( [1/(2([1/4]))] ) = − 2y = [(( − 2 )
^{2})/4] + ( − 2 ) = − 1 - Determine which direction the parabola opens

a > 0 opens upwards - Consider vertex and direction to determine maximum/minimum

( - 2, - 1) is a minimum

Find the maximum/minimum point of the function y = − 5x

^{2}+ 10x + 4- Find the vertex using x = − [b/2a]x = − ( [10/(2( − 5))] ) = 1y = − 5( 1 )
^{2}+ 10( 1 ) + 4 = 9 - Determine which direction the parabola opens

a < 0opens downwards - Consider vertex and direction to determine maximum/minimum

(1,9) is a maximum

Find the maximum/minimum point of the function y = [( − x

^{2})/20] + 6- Find the vertex using x = − [b/2a]x = − ( [0/(2( − [1/20]))] ) = 0y = [( − ( 0 )
^{2})/20] + 6 = 6 - Determine which direction the parabola opens

a < 0opens downwards - Consider vertex and direction to determine maximum/minimum

(0,6) is a maximum

Graph y = x

^{2}+ 4x + 4- Find the vertex using x = − [b/2a]x = − ( [4/2(1)] ) = − 2y = ( − 2 )
^{2}+ 4( − 2 ) + 4 = 0 - Make a table of points

x line − 2 − 5 − 4 0 1 y line 0 9 4 4 9

Graph y = − x

^{2}− 6x- Find the vertex using x = − [b/2a]x = − ( [( − 6)/(2( − 1))] ) = − 3y = − ( − 3 )
^{2}− 6( − 3 ) = 9 - Make a table of points

x line − 5 − 4 − 3 − 2 − 1 y line 5 8 9 8 5

Graph y = x

^{2}− 10x + 18- Find the vertex using x = − [b/2a]x = − ( [( − 10)/2(1)] ) = 5y = ( 5 )
^{2}− 10( 5 ) + 18 = − 7 - Make a table of points

x line 3 4 5 6 7 y line − 3 − 6 − 7 − 6 − 3

Solve y = x

^{2}+ 4x utilizing axis of symmetry- Find the vertex and line of symmetry using x = − [b/2a]x = − ( [4/2(1)] ) = − 2y = ( − 2 )
^{2}+ 4( − 2 ) = − 4 - Make a table of points

x line − 2 − 1 0 1 2 y line − 4 − 3 0 5 12

Graph to find x - intercepts

x = − 4,0

x = − 4,0

Solve y = − x

^{2}+ 16x − 55 utilizing axis of symmetry- Find the vertex and line of symmetry using x = − [b/2a]x = − ( [16/(2( − 1))] ) = 8y = − ( 8 )
^{2}+ 16( 8 ) − 55 = 9 - Make a table of points

x line 4 5 6 7 8 y line − 7 0 5 8 9

Graph to find x - intercepts

x = 5,11

x = 5,11

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Graphing Quadratic Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Parabolas 0:14
- Standard Form of Quadratic Function
- Examples
- Absolute Value of 'a'
- Parabolas That Open Upward 3:14
- Minimum
- Example
- Parabolas That Open Downward 6:57
- Example
- Maximum
- Vertex 9:53
- Example
- Axis of Symmetry 14:16
- Example
- Example 1: Graph the Quadratic 19:54
- Example 2: Graph the Quadratic 24:12
- Example 3: Vertex Maximum or Minimum 28:32
- Example 4: Axis of Symmetry 31:13

1 answer

Last reply by: Dr Carleen Eaton

Sun May 1, 2016 2:35 PM

Post by Kenosha Fox on April 28, 2016

I'm confused....in solving parabolas that open both upward and downward,do I always keep "a",which is sometimes 1 if there isn't a number in front of the "X", in the equation "ax+b...." positive?Weither or not if my number for "a" is positive or negative?

2 answers

Last reply by: Jessie Carrillo

Mon Feb 1, 2016 11:28 AM

Post by Jessie Carrillo on January 27, 2016

On example II. How did you get a negative Y value when you substituted -3 into Y=-x^2-4x-4?

1 answer

Last reply by: Dr Carleen Eaton

Sat Jul 27, 2013 10:36 AM

Post by Diana Del Angel on July 26, 2013

can you explain how you got the points to plot?

2 answers

Last reply by: Dr Carleen Eaton

Sat Jun 22, 2013 11:24 AM

Post by Taylor Wright on June 22, 2013

Can a parabola not open horizontally?

1 answer

Last reply by: Dr Carleen Eaton

Mon Nov 19, 2012 11:42 PM

Post by Erica Rapetti on November 19, 2012

I am confused how you got Y=-x* -2x+1

NOTE: * = the exponent 2