Roxee
Joseph
Paulo
Thang
a(x - h)² + ky = ax² + bx + c
a – width parabola
b – vertical shift
c – horizontal shift
h – vertical shift
k - horizontal
Graphs
• Important Parts of the
Graph
– Vertex
– Axis of Symmetry
– Zeros (Intercepts)
– Opening
Important Parts of the Graph
• Vertex
– The Min/Max point of
the Parabola
• Axis of Symmetry
– Imaginary line that
goes th...
How to Use a Graphing
Calculator
• Buttons to know:
– Y=
– Graph
– Trace
– Window
– Variable
What do the Buttons do?
• Y=
– Area where you input
equation(s) of the
graph
• Graph
– Shows the graph(s)
• Window
– Adjus...
How to Input a Graph on the
Calculator
• Click “Y=” button to
start
• Input your equation(s)
• Click the graph button
to v...
How to find the Vertex of your
Graph
• 2nd
Function > Trace
• Select Min or Max
depending on the
opening of the graph
• Se...
How to find the Zero’s of the
Graph
• 2nd
Function > Trace
• Select Zero’s
• Choose a point on the left side of a zero the...
Transformations
“By changing the equation, you can change the
shape, the direction it is pointing, and the location
of the...
General Parabola
 y= ax²

By adding a negative
sign before ‘a’ you
can change the
parabola from facing
north to facing s...
Shape of Parabola
y= 1x²y= 0.5x²
y= 2x²
In the equation y= ax², coefficient ‘a’
determines the shape of the parabola.
Maki...
Location of Parabola
In the equation
y= a (x-h) ²
‘h’ determines the
horizontal
movement of
the parabola
 If h > 0 then ...
Location of Parabola (pt.2)
In the equation
y= a (x-h) ² + k
‘k’ determines the
vertical
movement of
the parabola
 If k >...
Completing a Square
y= ax²+bx+c
Example One Example Two Example Three
y= x² +6x-7 y= 3x² +24x+21 y= 2x² +5x+2
= (x² +6x+_9...
Word Problems
Type One  good idea to draw a picture of some sort
A rectangular area is 600m. What are the dimensions of t...
Continued..
Type Two  make a table to help you
A company sells boots for $40, 600 people buy this product. For
every $10 ...
Continued..
Type Three  the equation is always already given in the question
A soccer ball is thrown up in the air with a...
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Quadratic Functions

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Transcript of "Quadratic Functions"

  1. 1. Roxee Joseph Paulo Thang
  2. 2. a(x - h)² + ky = ax² + bx + c a – width parabola b – vertical shift c – horizontal shift h – vertical shift k - horizontal
  3. 3. Graphs • Important Parts of the Graph – Vertex – Axis of Symmetry – Zeros (Intercepts) – Opening
  4. 4. Important Parts of the Graph • Vertex – The Min/Max point of the Parabola • Axis of Symmetry – Imaginary line that goes through the vertex. It splits the parabola into two mirror images • Zeros – Point(s) on the graph where it crosses the X- axis • Y-intercept – Point on the graph were it crosses the Y- axis • Opening
  5. 5. How to Use a Graphing Calculator • Buttons to know: – Y= – Graph – Trace – Window – Variable
  6. 6. What do the Buttons do? • Y= – Area where you input equation(s) of the graph • Graph – Shows the graph(s) • Window – Adjusts the graph’s min/max X and Y values • Trace – Able to run along the graph – Second Function (calculate): Is able to find the min/max value, zeros, etc. of graph • Variable – Easy access to variables
  7. 7. How to Input a Graph on the Calculator • Click “Y=” button to start • Input your equation(s) • Click the graph button to view your graph
  8. 8. How to find the Vertex of your Graph • 2nd Function > Trace • Select Min or Max depending on the opening of the graph • Select a point on the left side of the vertex then the right side of it, then take a guess at what it could be • Vertex will show up at the bottom of screen
  9. 9. How to find the Zero’s of the Graph • 2nd Function > Trace • Select Zero’s • Choose a point on the left side of a zero then a right (points must have different signs), lastly take a guess • Zero will show up at the bottom of screen • Usually needed to be done twice
  10. 10. Transformations “By changing the equation, you can change the shape, the direction it is pointing, and the location of the parabola”
  11. 11. General Parabola  y= ax²  By adding a negative sign before ‘a’ you can change the parabola from facing north to facing south y= -ax²   x= ay²  By adding a negative sign before ‘a’ you can change the parabola from facing east to facing west x= -ay² 
  12. 12. Shape of Parabola y= 1x²y= 0.5x² y= 2x² In the equation y= ax², coefficient ‘a’ determines the shape of the parabola. Making it wider, or skinnier. If coefficient ‘a’ is greater than 0 but less than 1, then the parabola will increase in width making it look wider If coefficient ‘a’ is greater than 1, then the parabola will decrease in width making it look skinnier
  13. 13. Location of Parabola In the equation y= a (x-h) ² ‘h’ determines the horizontal movement of the parabola  If h > 0 then  the parabola shifts to the right  If h < 0 then the parabola shifts to the left In the equation x= a (y-h) ² ‘h’ determines the vertical movement of the parabola  If h < 0 then the parabola shifts to the upwards  If h > 0 then the parabola shifts downwards  y= a (x-2) ² x= a (y-2) ²
  14. 14. Location of Parabola (pt.2) In the equation y= a (x-h) ² + k ‘k’ determines the vertical movement of the parabola  If k > 0 then  the parabola shifts upwards  If k < 0 then the parabola shifts downwards In the equation x= a (y-h) ² + k ‘k’ determines the horizontal movement of the parabola  If k < 0 then the parabola shifts to the left  If k > 0 then the parabola shifts to the right 
  15. 15. Completing a Square y= ax²+bx+c Example One Example Two Example Three y= x² +6x-7 y= 3x² +24x+21 y= 2x² +5x+2 = (x² +6x+_9_)-7-9 = 3(x² +8x+_16_)+21-48 = 2(x² +5/2x+_25/16_)+2-25/8 = (x+3)²-16 = 3(x+4) ²-27 = 2(x+5/4)²-9/8 -to find the final term of any of these equations: divide 2nd term by 2 then square as shown - if there is a coefficient, factor it out as shown above - before putting in final term, you must multiply it by the coefficient - remember, what you do on one side, you must do to the other side 6/2 = 3 3² = 9 factor 3 out 8/2 = 4 4² = 16 factor 2 out 5/2 divide by 2 = 5/4 5/4² = 25/16
  16. 16. Word Problems Type One  good idea to draw a picture of some sort A rectangular area is 600m. What are the dimensions of this area if there is a definite wall? A = lxw = (600-2w)(w) = 600w-2w² = -2w² +600w = -2(w²-300w+_22,500__)+45,000 = -2(w-150) ²+45, 000 W = 150 L = 600-2(150) = 300m W = 150m - remember, factor out coefficient first -divide second term by 2 then square to find the final term
  17. 17. Continued.. Type Two  make a table to help you A company sells boots for $40, 600 people buy this product. For every $10 increase, 60 fewer people buy the boots. What is the maximum revenue? Price: 40+10x # Sold: 600-60x 40+10(3) = 70 600-60(3)= 420 (40+10x)(600-60x) = 2400-2400x+6000-600² = -600² + 3600x +2400 = -600(x²-6x + _9__)+24000 + 5400 = -600(x-3) ² +29 400 X = 3 http://www.sheepskin-boots-and- slippers.com/images/discount-ugg-boots.jpg
  18. 18. Continued.. Type Three  the equation is always already given in the question A soccer ball is thrown up in the air with an initial velocity of 120m/s. Find the height of the ball and the time required with this equation: H = -5t² +120t + 4 h= -5t²+120t+4 = -5(t² -24t+ _144_)4+720 = -5(t-12) ² +724 t= 12 seconds h= 724 m http://www.albion.edu/imsports/

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