Quadratic Functions
- 2. a(x - h)² + ky = ax² + bx + c a – width parabola b – vertical shift c – horizontal shift h – vertical shift k - horizontal
- 3. Graphs • Important Parts of the Graph – Vertex – Axis of Symmetry – Zeros (Intercepts) – Opening
- 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. How to Use a Graphing Calculator • Buttons to know: – Y= – Graph – Trace – Window – Variable
- 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. 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. 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. 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. Transformations “By changing the equation, you can change the shape, the direction it is pointing, and the location of the parabola”
- 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. 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. 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. 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. 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. 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. 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. 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/