5-2
Standard Form of a Quadratic Function
TEKS FOCUS
VOCABULARY
ĚStandard form of a quadratic function –
TEKS (4)(D) Transform a quadratic function f(x) = ax2 + bx + c to the
form f(x) = a(x - h)2 + k to identify the different attributes of f(x).
The standard form of a quadratic
function is f(x) = ax2 + bx + c with a ≠ 0.
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental
math, estimation, and number sense as appropriate, to solve problems.
ĚNumber sense – the understanding of
what numbers mean and how they are
related
Additional TEKS (1)(A), (1)(F), (4)(B)
ESSENTIAL UNDERSTANDING
'PSBOZRVBESBUJDGVODUJPO f(x) = ax2 + bx + c,UIFWBMVFTPGabBOEcQSPWJEFLFZ
JOGPSNBUJPOBCPVUJUTHSBQI
Properties
Quadratic Function in Standard Form
r 5IFHSBQIPG f(x) = ax2 + bx + c,a ≠ 0,JTBQBSBCPMB
r *Ga 7 0,UIFQBSBCPMBPQFOTVQXBSE*Ga 6 0,UIFQBSBCPMBPQFOTEPXOXBSE
b
r 5IFBYJTPGTZNNFUSZJTUIFMJOFx = -2a
b
r 5IFxDPPSEJOBUFPGUIFWFSUFYJT - 2a
5IFyDPPSEJOBUFPGUIFWFSUFYJTUIFyWBMVF
b
( b)
PGUIFGVODUJPOGPSx = -2a ,PSy = f -2a r 5IFyJOUFSDFQUJT(0,c)
y ax 2 bx c, a 0
y
xb
2a
y ax 2 bx c,, a 0
y
b ,f b
(
2a
( 2a ))
(0, c)
(0, c)
O
(2ab , f (2ab ))
x
O
x
xb
2a
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159
Problem 1
P
TEKS Process Standard (1)(C)
Finding the Features of a Quadratic Function
Graphing Calculator What are the vertex, the axis of symmetry, the maximum or
minimum value, and the range of y = 2x2 + 8x − 2?
How can you use a
calculator to find
the features of a
quadratic function in
standard form?
Graph the function.
Then use the CALC and
TABLE features.
The vertex is (2, 10).
Range:
y 10.
Minimum
x 2
y 10
X
Y1
5
4
3
2
1
0
1
8
22
88
100
88
22
8
Notice the
symmetry of yvalues.
10 is the
minimum value.
Axis of symmetry
is x 2.
Problem
P
bl
2
TEKS Process Standard (1)(F)
Graphing a Function of the Form y = ax 2 + bx + c
What is the graph of y = x2 + 2x + 3?
Step 1 *EFOUJGZabBOEca = 1,b = 2,c = 3
b
Step 2
y
Step 2 5IFBYJTPGTZNNFUSZJTx = - 2a
How can you use the
axis of symmetry?
The entire curve on one
side of the axis is the
mirror image of the curve
on the other side.
6
x = - 2
2(1)
-JHIUMZTLFUDIUIFMJOFx = -1
Step 3 5IFxDPPSEJOBUFPGUIFWFSUFYJT
S
b
BMTP - 2a
,PS -1
5IFyDPPSEJOBUFJTy =
1MPUUIFWFSUFY( -1,2)
( -1)2
Step 4
4
2
O
+ 2( -1) + 3 = 2
Step 4 4
JODFc = 3,UIFyJOUFSDFQUJT(0,3)5IFSFGMFDUJPOPG(0,3)BDSPTT
x = -1JT( -2,3)1MPUCPUIQPJOUT
Step 5 a 7 0DPOGJSNTUIBUUIFHSBQIPQFOTVQXBSE%SBXBTNPPUIDVSWF
UISPVHIUIFQPJOUTZPVGPVOEJO4UFQTBOE
160
Lesson 5-2
Step 5
4
Standard Form of a Quadratic Function
x
2
4
Step 3
Problem 3
P
Converting Standard Form to Vertex Form
How do you find h, k,
and a?
Find the vertex. This gives
you h and k. The value
for a is the same in both
forms.
What is the vertex form of y = 2x2 + 10x + 7? Use the vertex form to find the
vertex,
the axis of symmetry, the maximum or minimum value, and the range.
v
y = 2x2 + 10x + 7
Identify a and b.
b
x = - 2a = -
Find the x-coordinate of the vertex.
10
2(2)
= -25
y = 2( -25)2 + 10( -25) + 7
Substitute x = -2.5 into the equation.
= -55
5IFWFSUFYJT( -25,-55)
y = a(x - h)2 + k
Write the vertex form.
y = 2[x - ( -25)]2 + ( -55)
Substitute a = 2, h = -2.5, k = -5.5.
y = 2(x + 25)2 - 55
Simplify.
5IFWFSUFYGPSNJTy = 2(x +
25)2
- 55
*OUIFQSPDFTTPGGJOEJOHUIFWFSUFYGPSNZPVBMSFBEZGPVOEUIFWFSUFY( -25,-55)
#FDBVTFUIFWFSUFYPGBQBSBCPMBJTPOUIFBYJTPGTZNNFUSZZPVDBOVTFUIF
xDPPSEJOBUFűPGUIFWFSUFYUPGJOEUIFBYJTPGTZNNFUSZx = -25
4JODFUIFWBMVFPGaJTQPTJUJWFZPVLOPXUIFQBSBCPMBPQFOTVQXBSE5IJTNFBOT
UIBUűUIFűQBSBCPMBűIBTBNJOJNVNWBMVFFRVBMUPUIFyWBMVFPGUIFWFSUFY -55
3FNFNCFSUIBUUIFSBOHFJTUIFTFUPGBMMyWBMVFTPGUIFGVODUJPO4JODFUIFGVODUJPO
IBTűBűNJOJNVNBU -55UIFSBOHFDPOTJTUTPGBMMSFBMOVNCFSTHSFBUFSUIBOPSFRVBM
UPű -55*OJOUFSWBMOPUBUJPOUIFSBOHFJT[ -55,∞)
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Problem 4
P
Interpreting a Quadratic Graph
Length of bridge above arch
STEM
Bridges The New River
Gorge Bridge in West Virginia
is the longest steel singlearch bridge in the United
States. You can model the
arch with the function
y = −0.000498x2 + 0.847x,
where x and y are in feet. How
high above the river is the
arch? How long is the section
of bridge above the arch?
How can you tell
that the quadratic
function has a
maximum value?
Since a 6 0, the graph
of the function opens
down. The function has a
maximum value.
516 ft
The height of the arch
above the support base
and the length of the
bridge above the arch
Step 1 'JOEUIFWFSUFYPGUIFBSDI
S
b
087
x = - 2a = -
≈ 850
2( - 000098)
y = -000098(850)2 + 087(850) ≈ 360
5IFWFSUFYJTBCPVU(850,360)
Step 2 'JOEUIFIFJHIUPGUIFBSDI
BCPWFJUTTVQQPSUT
400
Find the vertex. The y-coordinate is the
height of the arch above the support
base. The x-coordinate is half the
distance between the supports.
y
(850, 360)
300
200
100
O
x = 850
400
The
height is
360 ft.
5
A function that models the arch
and the vertical distance from the
base of the supports to the water
Height of arch
800
1200
5IFyDPPSEJOBUFPGUIFWFSUFYJTUIFIFJHIU PGUIFBSDIBCPWFJUT
TVQQPSUT5IFBSDIJTBCPVUGUBCPWFJUTTVQQPSUT
Step 3 'JOEUIFIFJHIUPGUIFBSDIBCPWFUIFSJWFS
5IFBSDIJTBCPVU360GU + 516GU = 876GUBCPWFUIFSJWFS
Step 4 'JOEUIFMFOHUIPGUIFCSJEHFBCPWFUIFBSDI
162
Lesson 5-2
IFxDPPSEJOBUFPGUIFWFSUFYJTIBMGUIFMFOHUIPGUIFCSJEHFBCPWFUIFBSDI
5
5IFűMFOHUIPGUIBUQBSUPGUIFCSJEHFJTBCPVU850GU + 850GU = 1700GU
Standard Form of a Quadratic Function
x
1600
HO
ME
RK
O
NLINE
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Identify the vertex, the axis of symmetry, the maximum or minimum value,
and the range of each parabola.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. y = x2 + 2x + 1
2. y = -x2 + 2x + 1
3. y = x2 + 4x + 1
4. y = -x2 + 2x + 5
5. y = 3x2 - 4x - 2
6. y = -2x2 - 3x + 4
7. y = 2x2 - 6x + 3
8. y = -x2 - x
9. y = 2x2 + 5
Graph each function.
10. y = x2 + 6x + 9
11. y = -x2 - 3x + 6
12. y = 2x2 + 4x
Write each function in vertex form and identify the different attributes of f (x).
13. f (x) = x2 - 4x + 6
14. f (x) = x2 + 2x + 5
15. f (x) = 4x2 + 7x
16. f (x) = 2x2 - 5x + 12
17. f (x) = -2x2 + 8x + 3
18. f (x) = 4x2 + 3x - 1
9
19. Apply Mathematics (1)(A) A model for a company’s revenue from selling a
software package is R = -2.5p2 + 500p, where p is the price in dollars of the
software. What price will maximize revenue? Find the maximum revenue.
Sketch each parabola using the given information.
20. vertex (3, 6), y-intercept 2
21. vertex ( -1, -4), y-intercept 3
22. vertex (0, 5), point (1, -2)
23. vertex (2, 3), point (6, 9)
24. Apply Mathematics (1)(A) A town is planning
a playground. It wants to fence in a rectangular
space using an existing wall. What is the
greatest area it can fence in using 100 ft of
donated fencing?
ℓ
100 – 2ℓ
25. Apply Mathematics (1)(A) Suppose you work
for a packaging company and are designing a box
that has a rectangular bottom with a perimeter of
36 cm. The box must be 4 cm high. What dimensions
give the maximum volume?
ℓ
For each function, the vertex of the function’s graph is given.
Find the unknown coefficients.
26. y = x2 + bx + c; (3, -4)
27. y = -3x2 + bx + c; (1, 0)
28. y = ax2 + 10x + c; ( -5, -27)
29. y = c - ax2 - 2x; ( -1, 3)
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30. Apply Mathematics (1)(A) 5IFIFJHIUPGBQSPKFDUJMFGJSFETUSBJHIUVQ
JOUIFBJSXJUIBOJOJUJBMWFMPDJUZPGGUTJTh = 6t - 16t 2 XIFSFhJT
IFJHIUJOGFFUBOEtJTUJNFJOTFDPOET5IFUBCMFSFQSFTFOUTUIFEBUB
GPSBOPUIFSQSPKFDUJMF8IJDIQSPKFDUJMFHPFTIJHIFS )PXNVDI
IJHIFS
31. "TUVEFOUTBZTUIBUUIFHSBQIPGy = ax2 + bx + cHFUTXJEFSBTa
JODSFBTFT
Time (t)
Height (h)
0.5
20
1
32
1.5
36
2
32
a. Evaluate Reasonableness (1)(B) 6TFFYBNQMFTUPTIPXUIBUUIF
TUVEFOUJTXSPOH
b. Connect Mathematical Ideas (1)(F) 4VNNBSJ[FUIFSFMBUJPOTIJQCFUXFFO 0 a 0 BOEUIFXJEUIPGUIFHSBQIPGy = ax2 + bx + c
For each function, find the y-intercept.
32. y = (x - 1)2 + 2
33. y = -3(x + 2)2 - 34. y = - 23(x - 9)2
35. 6TF UIF GVODUJPOT f (x) = x + 3 BOE g(x) = 12x2 + 2UPBOTXFSQBSUT(B) -(c)
a. 8IJDI GVODUJPO IBT B HSFBUFS SBUF PG DIBOHF GSPN x = 0 UP x = 1
b. 8IJDI GVODUJPO IBT B HSFBUFS SBUF PG DIBOHF GSPN x = 2 BOE x = 3
c. %PFT g(x) FWFS IBWF B HSFBUFS SBUF PG DIBOHF UIBO f (x) &YQMBJO
For each function, the vertex of the function’s graph is given. Find a and b.
36. y = ax2 + bx - 27; (2, -3)
37. y = ax2 + bx + 5; ( -1,)
38. y = ax2 + bx + 8; (2, -)
39. y = ax2 + bx; ( -3, 2)
40. 4LFUDI UIF QBSBCPMB XJUI BO BYJT PG TZNNFUSZ x = 2, yJOUFSDFQU BOE QPJOU TEXAS Test Practice
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41. 5IFUJNFJUUBLFT UPDIBMLBCBTFCBMMEJBNPOEWBSJFTEJSFDUMZXJUIUIFMFOHUIPG
UIFTJEFPGUIFEJBNPOE*GJUUBLFTNJOVUFTUPDIBMLBMJUUMFMFBHVFEJBNPOEXJUI
GUTJEFTIPXMPOHXJMMJUUBLFUPDIBMLBNBKPSMFBHVFCBTFCBMMEJBNPOEXJUI
GUűTJEFT
42. 8IBUJTUIFxWBMVFPGUIFWFSUFYPGUIFRVBESBUJDGVODUJPOy = -5x2 + 7
43. 4BSBIXPSLTBTBOBOOZBOEDIBSHFTEJGGFSFOUSBUFTGPSXPSLJOHEVSJOHUIFXFFLBOE
UIFXFFLFOE0OFXFFLTIFFBSOFEXPSLJOHIPVSTPGXIJDIIPVST
XFSFEVSJOHUIFXFFLFOE5IFGPMMPXJOHXFFLTIFFBSOFEXPSLJOHIPVST
PGXIJDIIPVSTXFSFEVSJOHUIFXFFLFOE8IBUEPFT4BSBIDIBSHFQFSIPVSJO
EPMMBSTGPSXPSLJOHEVSJOHUIFXFFL
164
Lesson 5-2
Standard Form of a Quadratic Function