โข WARMUP
(3,2) X=3
All Real Numbers
y โค 2
Reflected over x
Stretched vertically, right 3, up 2
(2,0) (4,0)(0 , -16)
< x < 3๐<๐<โ
VERTEX FORM: STANDARD FORM:
SKILL : CONVERT BETWEEN THE TWO Algebraic and Graphical Conversion
Vertex to Standard form โ Use the MATHStandard to Vertex form โ Use the Graphing Calculator
PRACTICE: Calculate/Break Down
FOIL!!First, Outer, Inner, Last
A third way to do this!!Combine like terms
x2 + 8x + 16
Distribute first factor into second factor
โ 6x + 9 + 9
x 4
x
4
x2 4x
4x 16
No matter which method you choose, put original equation into y1 and your answer into y2 and check to be sure the tables match!!!
Finally, a FOURTH way to do this, using the calculator.1) Put the equation you are given into the calculator, in y=2) Look at the table, and pick a few points.
Preferably, pick at least one point on either side of the vertex.3) Go to STAT Edit, and enter the points you chose into L1 (x-values) and L2 (y-values).4) Go to STAT Calc 5 and get your equation.5) As always, put both equations into y= and make sure that the tables match.
What does the graph look like?We know that the vertex is at (2,4) because itโs in vertex form.We know that the graph opens up, because itโs positiveSo, the graph looks roughly like this:
(๐โ๐ ) (๐ โ๐ )+๐FOIL METHOD
First, Outer, Inner, Last
DISTRIBUTE
x -2
x
-2
x2 -2x
-2x 4BOX SAME ANSWER EVERY TIME!
๐๐โ๐ ๐+๐(๐ ,๐) ๐=๐
๐จ๐๐ ๐น๐๐๐๐ต๐๐๐๐๐๐๐ โฅ๐
๐ด๐๐๐๐๐๐๐ต๐๐๐(๐ ,๐)๐<๐<โโโ<๐<๐
-2 FOIL METHOD-2First, Outer, Inner, Last
1) 1)
DISTRIBUTE) )
x -1
x
-1
x2 -x
-x 1BOX
โ๐ ๐๐+๐ ๐โ๐(๐ ,๐) ๐=๐
๐จ๐๐ ๐น๐๐๐๐ต๐๐๐๐๐๐๐ โค๐
๐ด๐๐๐๐๐๐
(๐ ,โ๐)
๐<๐<โโโ<๐<๐
(๐ ,๐)
This is already in standard formPut it into y=, and use 2nd Trace 4 to find the
vertex.The vertex is at (3,2)
We can also find the vertex by hand.The x-value of the vertex is found by
Once the x-value has been found, use it to find y.
If x = 3, then
Once again, the vertex is at (3,2)
(๐ ,๐) ๐=๐๐น๐๐๐๐๐๐ ๐๐๐๐ ๐ ;
๐ฝ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐ ,๐๐๐๐๐ ๐ ,๐๐๐
๐=โ๐ (๐โ๐ )๐+๐๐จ๐๐ ๐น๐๐๐๐ต๐๐๐๐๐๐๐ โค๐
๐ด๐๐๐๐๐๐
(๐ ,โ๐๐)
๐<๐<โโโ<๐<๐
(๐ ,๐ )(๐ ,๐)
โโโโ
(๐๐ ,
๐๐ ) ๐=
๐๐
๐น๐๐๐๐๐๐ ๐๐๐๐ ๐ ;๐ฝ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐ ,๐๐๐๐๐ ๐๐ ,๐๐
๐๐
๐=โ๐(๐โ๐๐ )๐+๐๐
๐จ๐๐ ๐น๐๐๐๐ต๐๐๐๐๐๐๐ โค๐๐
๐ด๐๐๐๐๐๐
(๐ ,๐)
๐๐<๐<โ
โโ<๐<๐๐
(๐ ,๐ )(๐๐ ,๐)
โโโโ
Find Parabola Fitting over Points
Estimation/ModelingQuadratic Regression and Calculations
1 -7 5
๐ฅ2โ7 ๐ฅ+5
-1 4 -7
โ๐ฅ2+4 ๐ฅโ7
2 -7 8
2 ๐ฅ2โ7 ๐ฅ+8
-17.07 110.73 -3
โ17.07 ๐ฅ2+110.73 ๐ฅโ3Enter equation in y= and look at the table for x = 6.
OR โ simply plug 6 into the equation by hand.
Either way, we get that the ball will be around 47 feet off the ground after 6 seconds.
Use the three points that we have to find the equation, then use the equation to find the two missing y-values.
Put 1, 2, and 3 into L1 and 25, 38, and 75 into L2STAT Calc 5 gets us the equation of
12 -23 36
12๐ฅ2โ23 ๐ฅ+36
Put into y=, then go 2nd Graph and look at the table for x=4 and x=5.
136 221
-.1 1 6
โ .1 ๐ฅ2+๐ฅ+6To find x = 8.5 on the graph, go to 2nd Window, and change TblStart= to 8.5.2nd Graph will show you 7.275
-.1 1 6
โ .1 ๐ฅ2+๐ฅ+6You can also use the STO button to find this. Type 8.5, STO, x. Then type in the equation (not in y=) to get your answer.Again, you will get 7.275
FIND THE VERTEX!! The y-value will be the lowest point, and the x-value will be the distance from the tower.To use the calculator, you have to play with your window to see the graph. Remember, the y-intercept is at 150. I used a range of 0 to 100 on the x-axis and from 0 to 165 on the y.By hand, use to find x, then plug that into the equation to find y.Either way, the vertex is (35,27.5)The cable gets within 27.5 feet above the roadway, and is 35 feet from the tower at that point.
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