~ Chapter 10 ~
Quadratic Equations and Functions
Lesson 10-1 Exploring Quadratic Graphs
Lesson 10-2 Quadratic Functions
Lesson 10-3 Finding & Estimating Square Roots
Lesson 10-4 Solving Quadratic Equations
Lesson 10-5 Factoring to Solve Quadratic Equations
Lesson 10-6 Completing the Square
Lesson 10-7 Using the Quadratic Formula
Lesson 10-8 Using the Discriminant
Lesson 10-9 Choosing a Linear, Quadratic, or
Exponential Model
Chapter Review
Exploring Quadratic Graphs
Cumulative Review Chap 1-9
Exploring Quadratic Graphs
Notes
Quadratic Function – a function in the form ax2 + bx + c, where a ≠ 0.
Examples ~ y = 2x2,
y = x2 -7,
y = x2 – x – 3
The graph of a quadratic function is a parabola…
The graph of y = x2 is
~>
~>
~>
A parabola can be folded so that the two sides match exactly. The
line that divides the parabola into two matched sides is called the axis
of symmetry.
The highest or lowest point of a parabola is called its vertex.
If a > 0 in y = ax2 + bx + c
~> The parabola opens upward
~> The vertex is the minimum point
If a < 0 in y = ax2 + bx + c
~> The parabola opens downward
~> The vertex is the maximum point
The vertex is identified as an ordered pair and as minimum or maximum…
Exploring Quadratic Graphs
Notes
Identify the vertex of each graph and tell whether it is a minimum or a
maximum.
Graphing y = ax2
(1) Make a table of values.
(2) Graph the points.
(3) Find the corresponding points on the other side of the axis of
symmetry.
Graph f(x) = -2x2
The value of a affects the width of the parabola as well as the
direction it opens.
You can order quadratic functions by their widths.
Order y = x2, y = ½x2, and y = -2x2 from widest to narrowest…
Graphing y = ax2 + c
The value of c translates the vertex of the graph up (+) or down (-).
Exploring Quadratic Graphs
Notes
Graph y = 2x2 + 3
Graph y = -1/2x2 – 4
In summary…
(1) The coefficient of x2, a, determines the width and whether the
parabola points upward (+) or downward (-).
(2) The constant, c, determines the vertex location above or below 0.
(3) Ordering quadratic graphs by width, the smaller the coefficient, a,
of x2, the wider the graph.
Exploring Quadratic Graphs
Homework
Homework – Practice 10-1
#1-26
Quadratic Functions
Practice 10-1
Quadratic Functions
Practice 10-1
Quadratic Functions
Practice 10-1
Quadratic Functions
Notes
Graphing y = ax2 + bx + c
y = 2x2 + 2x
(1) Find the axis of symmetry… x = -b/2a Then find the y
coordinate. These are the coordinates of the vertex.
(2) Find two other points on the graph.
(3) Reflect those two points over the axis of symmetry. Draw the parabola.
Graph 2x2 + 2x
Graph f(x) = x2 – 6x + 9
Graphing Quadratic Inequalities
y ≤ x2 + 2x – 5
Graph the boundary curve…
Shade the area below the curve because it is less than or equal to.
Quadratic Functions
Notes
Graph y > x2 + x + 1
Quadratic Functions
Homework
Homework ~ Practice 10-2 even
Finding & Estimating Square Roots
Practice 10-2
Finding & Estimating Square Roots
Practice 10-2
Finding & Estimating Square Roots
Practice 10-2
Finding & Estimating Square Roots
Practice 10-2
Finding & Estimating Square Roots
Notes
Finding Square Roots
Every positive number has two square roots… The square root of 16 = 4 and -4
or ± 4.
√25 means the positive or principal square root of 25 which is 5. -√25 means
the negative square root of 25 which is -5. You can use ± to represent both
square roots.
Simplifying Square Root Expressions
√64 =
-√100 =
±√49 =
√1/25 =
- √121 =
Rational & Irrational Square Roots
Rational square roots have a terminating or repeating decimal…
Irrational square roots have decimals that do not repeat.
±√81 =
√8 =
-√225 =
√75 =
±√1/4 =
Estimating Square Roots
You can estimate square roots by using perfect squares. Estimation places the
square root between two consecutive integers.
Finding & Estimating Square Roots
Example - √18.5
Notes
√16 < √18.5 < √25 so…
4 < √18.5 < 5
so √18.5 is between 4 and 5.
Your turn -√105 is between what two consecutive integers?
Approximating Square Roots with a Calculator
Find √18.5 to the nearest hundredth…
Find √17.81 to the nearest hundredth
Find -√203 to the nearest hundredth
Finding & Estimating Square Roots
Homework
Homework – Practice 10-3 odd
Solving Quadratic Equations
Practice 10-3
Solving Quadratic Equations
Notes
Solving Quadratic Equations by Graphing
A quadratic equation is an equation that can be written in the form…
ax2 + bx+ c = 0, where a ≠ 0. This is the standard form of a quadratic
equation.
Quadratic equations can have two, one, or no real-number solutions.
Algebra I focusing only on real-number solutions.
Solve by graphing… x2 – 4
What about
(The solution(s) are the x-intercepts)
x2 = 0 ?
x2 – 1 = 0
2x2 + 4 = 0
x2 – 16 = -16
Solving Quadratic Equations Using Square Roots
To solve an equation in the form x2 = a; find the square roots of both sides.
Solving Quadratic Equations
Notes
t2 – 25 = 0
t2 = 25
t=±5
Find the solution(s) 3n2 + 12 = 12
Try…
2g2 + 32 = 0
Factoring can also be used to solve the quadratic equation…
x2 - 9 = 0
(x + 3) (x – 3) = 0
(x + 3) = 0
x = -3
or (x – 3) = 0
x=3
Solutions ±3
Solving Quadratic Equations
Homework
Homework – Practice 10-4
odd
Factoring to Solve Quadratic Equations
Practice 10-4
~ Chapter 10 ~
Chapter Review
Chapter 10 Review Part 1
Chapter Review
Chapter 10 Review Part 1
Chapter Review
Chapter 10 Review Part 1
Chapter Review
Factoring to Solve Quadratic Equations
Notes
Using the Zero-Product Property
If ab = 0, then a = 0 or b = 0
Solve (x + 7) (x – 4) = 0
So…
x + 7 = 0 or x – 4 = 0
x = -7
or
x=4
Your turn… Solve (3y – 5) (y – 2) = 0
Solving by Factoring
x2 – 8 x – 48 = 0
(x – 12) (x + 4) = 0
x – 12 = 0 or
x = 12
or
x+4=0
x = -4
Your turn… x2 + x – 12 = 0
x = -4 or x = 3
Factoring to Solve Quadratic Equations
Notes
2x2 – 5x = 88
2x2 – 5x – 88 = 0
Your turn…
Solve x2 – 12x = -36
x2 – 12x + 36 = 0
(x - 6)2 = 0
x=6
Factoring to Solve Quadratic Equations
Homework
~ Homework ~
Practice 10-5 even
Completing the Square
Practice 10-5
Using the Quadratic Formula
Notes
Using the Quadratic Formula
If ax2 + bx + c = 0, and a ≠ 0, then…
x = -b ±
b2 – 4ac
2a
Make sure your quadratic equation is in standard form…
Solve x2 + 6 = 5x
x2 – 5x + 6 = 0
x = - (-5) ±
x=5±
(-5)2 – 4(1)(6)
2(1)
25 – 24
2
x = 5 ± √ 1 = 5 + 1 or 5 – 1
2
2
2
= 6
2
or 4
2
=
3 or 2
Using the Quadratic Formula
Notes
Your turn…
Solve using the quadratic formula
x2 – 4x = 117
x2 – 2x – 8 = 0
Finding Approximate Solutions
2x2 + 4x – 7 = 0
x = - (4) ± (4)2 – 4(2)(-7)
2(2)
= -4 ±
16 – (-56)
4
x = -4 + √72 or -4 - √72 ≈ -4 + 8.49 or -4 - 8.49 ≈ 1.12 or -3.12
4
4
4
4
Your turn…
7x2 – 2x – 8 = 0
Using the Quadratic Formula
Homework
Homework – Practice 10-7 even
Using the Discriminant
Practice 10-7
Using the Discriminant
Notes
Number of Real Solutions of a Quadratic Equation
Discriminant – The expression under the radical in the quadratic formula.
(b2 – 4ac) The discriminant can be used to determine how many solutions a
quadratic equation has before you solve it…
If b2 – 4ac > 0, there are 2 solutions
If b2 – 4ac = 0, there is 1 solution
If b2 – 4ac < 0, there are no solutions
Using the Discriminant
Find the number of solutions for x2 – 2x – 3
b2 – 4ac = (-2)2 – 4(1)(-3)
4 – (-12) = 4 + 12 = 16 > 0 , so there are 2 solutions.
Your turn… Find the number of solutions for 3x2 – 4x – 7
Find the number of solutions for 5x2 + 8 = 2x
Using the Discriminant
Homework
Homework – Practice 10-8 odd
&
Chapter 10 Review Part 2
Using the Discriminant
Homework
~ Chapter 10 ~
Chapter Review
~ Chapter 10 ~
Chapter Review