Chapter 8 – Quadratic Functions and Equations Class Notes

Identifying Quadratic Functions Lesson 8.1

Identifying Quadratic Functions A quadratic function does not have constant first differences but constant second differences Standard form of quadratic functions Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐

Graphing Quadratic Functions by Using a Table of Values Make a table of values. Choose values of x and use them to find values of y X Y=2 𝑥 2 -2 8 -1 2 1

Graphing Quadratic Functions by Using a Table of Values Graph the points. Then connect the points with a smooth curve.

Identifying the Direction of a Parabola When a quadratic function is written in the form Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐, the value of a determines the direction a parabola opens. a > 0 – Parabola opens UPWARD (U-Shaped) a < 0 – Parabola opens DOWNWARD (Rainbow Shaped)

Identifying the Vertex and the Minimum or Maximum The highest or lowest point on a parabola is the vertex If a > 0, the parabola opens upward, and the y-value of the vertex is the minimum value of the function If a < 0, the parabola opens downward, and the y-value of the vertex is the maximum value of the function

Finding Domain and Range Unless a specific domain is given, you may assume that the domain of a quadratic function is all real numbers.

Characteristics of Quadratic Functions Lesson 8.2

Finding Zeros of Quadratic Functions From Graphs A zero of a function is an x-value that makes the function equal to 0. A zero function is the same as an x-intercept of a function. A quadratic function may have one, two or no zeros. Identify the zeros below

ANSWER 1. 2 and -1 2. 1 3. No Zeros

Finding the Axis of Symmetry by Using Zeros A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry ONE ZERO If a function has one zero, use the x-coordinates of the vertex to find the axis of symmetry TWO ZEROS If a function has two zeros, use the average of the two zeros to find the axis of symmetry

Finding the Axis of Symmetry by Using Zeros Identify the axis of symmetry for each graph

ANSWER 1. -3 2. 1

Finding the Axis of Symmetry by Using the Formula If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry.

Finding the Axis of Symmetry by Using the Formula Identify the axis of symmetry for each equation by using the formula

ANSWER -1/4

Finding the Vertex of a Parabola Step 1 To find the x-coordinate of the vertex, find the axis of symmetry by using zeros or the formula Step 2 To find the corresponding y-coordinate, substitute the x-coordinate of the vertex into the function Step 3 Write the vertex as an ordered pair

Finding the Vertex of a Parabola (Example)

Finding the Vertex of a Parabola Find the vertex by using the Axis of Symmetry Formula

ANSWER (2, -14)

Additional Practice Workbook Page 425 DUE Tomorrow when class begins (Put in Class Folder upon entering class) If you complete it in class, place it in your Class Folder on your way out

Graphing Quadratic Functions Lesson 8.3

Graphing a Quadratic Function Recall Standard Form of Quadratic Function Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐 Remember that when x=o, y=c The y-intercept of a quadratic function is c.

Graphing a Quadratic Function Step 1: Find the axis of symmetry (Use Formula from 8.2) Step 2: Find the vertex (Substitute your x-coordinate into your function and solve for y Step 3: Find your y-intercept (Identify c)

Graphing a Quadratic Function Step 4: Find two more points on the same side of the axis of symmetry as the point containing the y-intercept (Choose values less than your axis of symmetry) Substitute x-coordinates Step 5: Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. Reflect the points across the axis of symmetry and connect points with smooth curve

Graphing a Quadratic Function Graph quadratic function and label your steps 1-5 on your whiteboard and raise when you are finished

ANSWER

Additional Practice Workbook page 429 DUE Tomorrow (Place in class folder as you walk into class)

Warm Up 2 𝑥 2 +5𝑥+2 Find a Find b Find − 𝑏 2𝑎 Find the Axis of Symmetry Find the Vertex

Transforming Quadratic Functions Lesson 8.4

Comparing Widths of Parabolas The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola

Comparing Widths of Parabolas Order the functions in order from most narrow to the widest

Comparing Graphs of Quadratic Functions The value of c makes these graphs look different

Comparing Graphs of Quadratic Functions Two Methods Comparing the graphs Comparing the functions

Additional Practice Workbook page 435 DUE Friday (Place in class folder as you walk into class)

TEST Get ready and study for test on Lesson 8.1 – 8.4

Solving Quadratic Equations by Graphing Lesson 8.5

Solving Quadratic Equations by Graphing

Solving Quadratic Equations by Graphing 2𝑥 2 −2=0 Write the Related function 2𝑥 2 −2=𝑦 𝑜𝑟 𝑦= 2𝑥 2 +0𝑥−2 Graph the function Axis of Symmetry = 0 Vertex = (0,-2) Two other points = (1,0) and (2,6) Graph the points and reflect them across the axis of symmetry Find the zeros The zeros appear to be -1 and 1

Solving Quadratic Equations by Graphing

Solving Quadratic Equations by Factoring Lesson 8.6

Using the Zero Product Property

Using the Zero Product Property (x – 3)(x + 7) = 0 Use the zero property x – 3 = 0 …. x = 3 x + 7 = 0 …. x = -7 The solutions are 3 and -7 Can always check your work by plugging each solution for x into the original equation

Solving Quadratic Equations by Factoring If a quadratic equation is written in standard form, you may need to factor before using the Zero Product Property 𝑥 2 +7𝑥+10=0 𝑥+5 𝑥+2 =0 𝑥+5=0 …𝑥=−5 𝑥+2=0 …𝑥=−2 The solutions are -5 and -2

Solving Quadratic Equations by Factoring −2𝑥 2 =18−12𝑥 −2𝑥 2 +12𝑥−18=0 −2 𝑥 2 −6𝑥+9 =0 −2 𝑥−3 𝑥−3 =0 −2≠0, 𝑥=3

Additional Practice Workbook page 451

Solving Quadratic Equations by Using Square Roots Lesson 8.7

Using Square Roots to Solve 𝑥 2 =𝑎

Using Square Roots to Solve 𝑥 2 =𝑎 When you take the square root of a positive real number and the sign of the square root is not indicated, you must find both the positive and negative square root. This is indicated by ±√ 𝑥 2 =16 𝑥=± 16 𝑥=±4 The solutions are 4 and -4

Using Square Roots to Solve 𝑥 2 =𝑎 𝑥 2 =−4 𝑥=± −4 There is no real number whose square is negative There is no real solution

Using Square Roots to Solve Quadratic Equations If necessary, use inverse operations to isolate the squared part of a quadratic equation before taking the square root of both sides 𝑥 2 +5=5 𝑥 2 =0 𝑥=± 0 =0 The solution is 0

Using Square Roots to Solve Quadratic Equations 4𝑥 2 −25=0 4𝑥 2 =25 𝑥 2 = 25 4 𝑥=± 25 4 𝑥=± 5 2 The solutions are 5 2 𝑎𝑛𝑑 − 5 2

Additional Practice Workbook Page 457 Finish Project Standards HRW DUE 2/20

Completing the Square Lesson 8.8

Completing the Square When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. An expression in the form 𝑥 2 +𝑏𝑥 is not a perfect square. However, you can use the relationship shown above to add a term to 𝑥 2 +𝑏𝑥 to form a trinomial that is a perfect square. This is called completing the square.

Completing the Square 𝑥 2 +10𝑥+ 𝑥 2 +10𝑥 10 2 2 = 5 2 =25 𝑥 2 +10𝑥+25

Solving a Quadratic Equation by Completing the Square

Solving 𝑥 2 +𝑏𝑥=𝑐 by Completing the Square 𝑥 2 +14𝑥=15 14 2 2 = 7 2 =49 𝑥 2 +14𝑥+49=15+49 𝑥+7 2 =64 𝑥+7=±8 𝑥+7=8 or 𝑥+7=−8 𝑥=−1 𝑜𝑟 𝑥=−15

Solving 𝑎𝑥 2 +𝑏𝑥=𝑐 by Completing the Square

Additional Practice Guided Practice p. 579 #’s 2-32 even Test Thursday

The Quadratic Formula and the Discriminant Lesson 8.9

Using the Quadratic Formula

Using the Quadratic Formula

Using the Quadratic Formula to Estimate Solutions

Using the Discriminant If the quadratic equation is in standard form, the discriminant of a quadratic equation is 𝑏 2 −4𝑎𝑐, the part of the equation under the radical sign

Using the Discriminant

Using the Discriminant

Solving Using Different Methods Factoring Completing the Square Using the Quadratic Formula

Additional Practice Workbook page 477

Nonlinear Systems Lesson 8.10

Solving a Nonlinear System by Graphing A nonlinear system of equations is a system in which at least one of the equations is nonlinear

Solving a Nonlinear System by Graphing

Solving a Nonlinear System by Substitution

Solving a Nonlinear System by Elimination

Additional Practice Workbook page 485 Quiz Tuesday 2/20 on 8.5 – 8.10