Chapter 3.1 – Quadratic Functions and Models - Pablo...
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Transcript of Chapter 3.1 – Quadratic Functions and Models - Pablo...
Chapter 3.1 – Quadratic Functions and Models
Chapter 3.1 Quadratic Functions and Models
Recall the graph of the parent square function.
Vertical Stretching of Graphs Consider the graphs of 2)( xxf = and 22)( xxg = . A table of values is given along with the graphs of each function. x f(x) g(x) -2 4 8 -1 1 2 0 0 0 1 1 2 2 4 8 Note that the graph of g(x) is the same graph of f(x) shifted stretched vertically.
Vertical Stretching The graph of 2)( axxg = is a parabola with vertex at the origin that opens up if a is positive and down if a is negative. The width of the graph of 2)( axxg = is determined by the magnitude of a. The graph of 2)( axxg = is narrower than
2)( xxf = if 1>a and is wider than 2)( xxf = if 1<a .
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A function f is a quadratic function if
cbxaxxf ++= 2)( where a,b, and c are real numbers and 0≠a .
Chapter 3.1 – Quadratic Functions and Models
Sample Problem: Sketch the graph of
a. 2)5()( 2 −−= xxg b. 3)1(2)( 2 ++−= xxh Solution: a. Note that this is a parabola whose vertex has been shifted 5 units to the right and 2 units down. No vertical shrinking has taken place. The graph is shown in purple below. b. Note that this is a parabola that opens down and is narrower than your parent function. Additionally, the vertex has been shifted 1 unit to the left and 3 units up. The graph is shown in green below. Student Practice :
Graphing Techniques and Quadratic Functions Any quadratic function can be written in the form
khxaxF +−= 2)()(
The graph of )(xF is the same as the graph of 2)( axxg = translated h units horizontally (to the right if h is positive and to the left if h is negative) and translated vertically (up if k is positive and down if k is negative). It is a parabola with vertex (h,k) and axis of symmetry x = h.
2)5()( 2 −−= xxg 3)1(2)( 2 −+−= xxh
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Chapter 3.1 – Quadratic Functions and Models
Completing the Square The method of completing the square can be used to transform any quadratic function into the form shown above, though it is not always the most practical way to graph a quadratic function. Consider the example: Sample Problem: Use completing the square to write the quadratic function in the form khxaxF +−= 2)()( , then sketch the graph.
a. 24)( 2 −−= xxxf b. 263)( 2 −+−= xxxh
Solution: a. 6)2(
244424)(
2
2
2
−−=
−−+−=
−−=
xxxxxxf
b.
1)1(323)12(3
2)2(3263)(
2
2
2
2
+−−=
−++−−=
−−−=
−+−=
xxxxx
xxxh
Graph has vertex ( 2, -6) and opens up. Graph has vertex ( 1,1 ) and opens down, narrow.
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24)( 2 −−= xxxf
263)( 2 −+−= xxxh
Chapter 3.1 – Quadratic Functions and Models
Student Practice: Complete the square for each quadratic function and sketch it’s graph. 1. 36)( 2 −+= xxxf 2. 582)( 2 −−−= xxxg The Vertex Formula An alternative method of graphing a quadratic function is to compute the vertex without completing the square. This can be done using the following:
Sample Problem: Find the vertex of each quadratic function.
a. 24)( 2 −−= xxxf b. 263)( 2 −+−= xxxh Solution:
a. Since a = 1, b = -4, and c = -2, Since a = -3, b = 6, and c = -2
62)2(4)2()2(
224
)1(2)4(
2 −=−−==
==−−
=
fk
h
12)1(6)1(3)1(
166
)3(2)6(
2 =−+−==
=−−
=−
−=
fk
h
Vertex ( 2 , -6 ) Vertex ( 1 , 1 )
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Vertex Formula The vertex for any quadratic function
cbxaxxf ++= 2)(
can be found using abh
2−
= and )(hfk = to get the vertex of the parabola ),( kh .
Chapter 3.1 – Quadratic Functions and Models
Student Practice: Graph using vertex formula and y intercepts. 1. 36)( 2 −+= xxxf 2. 582)( 2 −−−= xxxg
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Chapter 3.1 – Quadratic Functions and Models
Problems Involving Projectiles Sample Problem: Neglecting air resistance, the height h of a model rocket launched from the ground into the air with initial velocity of 192 ft/sec after t seconds can be found using the equation
tth 19216 2 +−= a. What is the maximum height achieved by the rocket? (Hint: Simply find the vertex)
