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Transcript
Chapter 9.1 & 9.2
Common Core – F.IF.7.a, A.CED.2, F.IF.4, F.IF.5,
F.IF.8a, F.IF.9, & F.BF.3 Graph linear and
quadratic functions and show intercepts,
maxima, and minima.
Objective – To graph quadratic functions of the
form y = ax2, y = ax2 + c and y = ax2 + bx +c .
Ch 9.1 – 9.2 Notes
Quadratic Graphs
y = aπ‘₯ 2 + bx + c
If a is positive it cups up
If a is negative it cups down
To find the vertex
βˆ’π‘
(
2π‘Ž
, plug it in)
Axis of symmetry x = the x-coordinate of the vertex
If you factor the quadratic it will tell you were you
cross the x-axis.
Chapter 9.1
Solving Quadratic Equations
When you solve the Quadratic it tells you where
you cross the x-axis.
1) x = 2 & x = 3 (Crosses the x-axis in 2 places)
2) x = -4 & x = -4 (Crosses the x-axis in 1 place)
3) Not factorable (Doesn’t cross the x-axis)
Chapter 9.3
Common Core – A.REI.4.b, N.Q.2, A.APR.3,
A.CED.1, & A.CED.4 Solve quadratic equations
by inspection, taking square roots.
Objective – To solve quadratic equations by
graphing and using square roots.
Ch 9.3 Notes
Solving Quadratic Equations
When you solve the Quadratic it tells you where
you cross the x-axis.
1) x = 2 & x = 3 (Crosses the x-axis in 2 places)
2) x = -4 & x = -4 (Crosses the x-axis in 1 place)
3) Not factorable (Doesn’t cross the x-axis)
Chapter 9.4
Common Core – A.REI.4.b, A.SSE.3.a, A.CED.1,
& A.F.IF.8a Solve quadratic equations by
factoring.
Objective – To solve quadratic equations by
factoring.
Ch 9.4 Notes
Solve by Factoring
A * B = 0 either A = 0 or B = 0
Examples
1) (4x + 1)(x – 2) = 0
2) π‘₯ 2 + 8π‘₯ + 15 = 0
Examples
3) 4π‘₯ 2 - 21x = 18
4) You are constructing a frame for a 17 inch by
11 inch photo. You want the frame to be the
same width all the way around and the total
area of the frame and photo to be 315 𝑖𝑛2 .
What is the outer dimensions of the frame?
Chapter 9.5
Common Core – A.REI.4.a, N.Q.3, A.SSE.1.a,
A.SSE.1.b, A.SSE.3.b., A.CED.1, A.REI.1,
A.REI.4.b, & A.F.IF.8a Use the method of
completing the square to transform any
quadratic equation in s into an equation of the
form (x – p)2 = q
Objective – To solve quadratic equations by
completing the square.
Ch 9.5 Notes
Completing the Square
1) Make sure the leading coefficient is equal to 1
(a = 1)
2) Move the constant term c to the right of the
equal sign. (move c.)
3) Take b and divide by two, square it and add it
𝑏 2
to both sides of the equation. ( )
2
4) Factor and Solve the Equation
Examples
1) What is the value of c such that π‘₯ 2 βˆ’ 16π‘₯ + 𝑐
is a perfect-square trinomial?
2) Solve π‘₯ 2 βˆ’ 14π‘₯ + 16 = 0
Examples
3) Solve 2π‘₯ 2 +12π‘₯ + 16 = 0
4) Solve 3π‘₯ 2 + 36π‘₯ + 102 = 0
Chapter 9.6
Common Core – A.REI.4.a, N.Q.3, A.CED.1, &
A.REI.4.b Use the method of completing the
square to transform any quadratic equation in s
into an equation of the form (x – p)2 = q… Derive
the quadratic formula from this form.
Objectives – To solve quadratic equations by
using the quadratic formula. To find the number
of solutions of a quadratic equation.
Ch 9.6 Notes
Quadratic Formula
aπ‘₯ 2 + 𝑏π‘₯ + 𝑐 = 0
x=
Example
1) π‘₯ 2 - 8 = 2x
βˆ’π‘± 𝑏2 βˆ’4π‘Žπ‘
2π‘Ž
Examples
2) 2π‘₯ 2 + 7π‘₯ βˆ’ 15 = 0
Examples
3) 3π‘₯ 2 - 17x + 11 = 0
aπ‘₯ 2 + 𝑏π‘₯ + 𝑐 = 0
x=
βˆ’π‘± 𝑏2 βˆ’4π‘Žπ‘
2π‘Ž
𝑏 2 - 4ac is called the discriminate
𝑏 2 - 4ac > 0 (There are 2 real solutions)
𝑏 2 - 4ac = 0 (There is 1 real solutions)
𝑏 2 - 4ac < 0 (There is no solution)
Find the number of real-number solutions of
each equation.
1) 3π‘₯ 2 βˆ’ 6π‘₯ + 9 = 0
2) 2π‘₯ 2 βˆ’ 15 = 0
3) 4π‘₯ 2 βˆ’ 20π‘₯ + 25 = 0
Chapter 9.7
Common Core – F.LE.a, F.IF.4, F.LE.2, F.LE.3 &
S.ID.6a Prove that linear functions grow by
equal differences…and that exponential
functions grow by equal factors over equal
intervals.
Objective – To choose a linear, quadratic, or
exponential model for data.
Ch 9.7 Notes
Linear, Quadratic, Exponential Models
y = mx + b (Linear)
y = aπ‘₯ 2 + bx + c (Quadratic)
y = a * 𝑏 π‘₯ (Exponential)