Finding the axis of symmetry, vertex, and roots of a parabola
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- Erika Fitzgerald
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1 Finding the axis of symmetry, vertex, and roots of a parabola 1) Find the Axis of Symmetry of y = x 2-4x + 3 (The AOS is the vertical line that splits the parabola in 2 equal parts) Axis of Symmetry Axis of Symmetry: x = -b. x = -(-4). x = 4. x = 2 A.O.S. x =2 2a 2(1) 2 [a is the coefficient of x 2 (1) and b is the coef. of x (-4)] a = 1 b = -4 2) Find the vertex of y = x 2 + 4x + 3 (The vertex is the turning point) Finding the vertex: If the axis of symm. is an integer, then you can find the ordered pair for the vertex in the table. If not, we will use the calculator. 2 nd CALC 1 (for value) Type in the value for x (x= 2) and hit ENTER The ordered pair for the vertex will appear on the bottom of the screen. (2,-1) 3) Find the roots of y = x 2 + 4x + 3 (The roots are where the parabola crosses the x-axis) Finding the roots: The roots are where the equation or y = 0. (Where it crosses the x-axis). This is why we put y = 0 into the calculator. The x-axis is y = 0. There are usually 2 roots. To find the first: 2 nd CALC 5 (for intersect) ENTER ENTER ENTER The ordered pair will be at the bottom. (1,0) Root #1: (1, 0) To find the 2 nd root: 2 nd CALC 5 (for intersect) DO NOT HIT ENTER!!!!! You must move the cursor using the blue left and right arrows (not up and down) close to the 2 nd root. Once close: ENTER ENTER ENTER The ordered pair will be at the bottom. (3,0) Root #2: (3,0) 1
2 1) Find the Axis of Symmetry of y = -3x 2 + x + 2 (The AOS is the vertical line that splits the parabola in 2 equal parts) Axis of Symmetry Axis of Symmetry: x = -b. x = -(1). x = A.O.S.= x = 2a 2(-3) [a is the coefficient of x 2 (-3)and b is the coef. of x (1)] 2) Find the vertex of y = -3x 2 + x + 2 (The vertex is the turning point) Finding the vertex: If the axis of symm. is an integer, then you can find the ordered pair for the vertex in the table. If not, we will use the calculator. 2 nd CALC 1 (for value) Type in the value for x= ( ) and hit ENTER The ordered pair for the vertex will appear on the bottom of the screen. ( , ) 3) Find the roots of y = 3x 2 + x + 2 (The roots are where the parabola crosses the x-axis) Finding the roots: The roots are where the equation or y = 0. (Where it crosses the x-axis). This is why we put y = 0 into the calculator. The x-axis is y = 0. There are usually 2 roots. To find the first: 2 nd CALC 5 (for intersect) ENTER ENTER ENTER The ordered pair will be at the bottom. Root #1: (, 0) To find the 2 nd root: 2 nd CALC 5 (for intersect) DO NOT HIT ENTER!!!!! You must move the cursor using the blue left and right arrows (not up and down) close to the 2 nd root. Once close: ENTER ENTER ENTER The ordered pair will be at the bottom. Root #2: (1,0) 2
3 For each parabola find the axis of symmetry, vertex, and roots: DO NOT ROUND!! 1) y = x 2 + 2x 8 2) y = x 2 4x 5 AOS: x = AOS: Roots: and Roots: and 3) y = -x 2 + x 6 4) y = -x 2 3x + 4 AOS: x = AOS: Roots: and Roots: and 3
4 For 5-8: Round all answers to the nearest hundredth: 5) y = 3x 2 + 2x 4 6) y = -2x 2 5x + 1 AOS: x = AOS: Roots: and Roots: and 7) y = -4x 2 + 5x 8) y = x 2 8 (be careful what s b equal to?) AOS: x = AOS: Roots: and Roots: and 4
5 1) a) Graph: y = -x 2-2x + 8 2) a) Graph: y = 2x 2 +2x 2 b) Is the vertex a minimum or maximum? b) Is the vertex a min. or max.? c) Find the axis of symmetry. c) Find the axis of symmetry. d) Find the vertex. d) Find the vertex. e) Find the roots. e) Find the roots. 5
6 3) a) Graph: y = x 2 3x - 6 4) a) Graph: y = -3x 2 + x + 2 b) Is the vertex a minimum or maximum? b) Is the vertex a min. or max.? c) Find the axis of symmetry. c) Find the axis of symmetry. d) Find the vertex. d) Find the vertex. e) Find the roots. e) Find the roots. 6
7 5) a) Graph: y = 2x 2 + 5x - 3 6) a) Graph: y = - 3x 2 4x + 3 b) Is the vertex a minimum or maximum? b) Is the vertex a min. or max.? c) Find the axis of symmetry. c) Find the axis of symmetry. d) Find the vertex. d) Find the vertex. e) Find the roots. e) Find the roots. 7
8 7) a) Graph: y = -2x 2 + 5x 8) a) Graph: y = x 2 7x + 6 b) Is the vertex a minimum or maximum? b) Is the vertex a min. or max.? c) Find the axis of symmetry. c) Find the axis of symmetry. d) Find the vertex. d) Find the vertex. e) Find the roots. e) Find the roots. 8
9 1) Solve the system graphically and check: y = -x 2 + 5x + 2 y + 10 = 3(x + 3) 9
10 2) Solve the system graphically and check: y = 2x 2 6x 20x + 5y = 20 10
11 3) Solve the system graphically and check: y = -x 2 x + 6 y = -3x
12 4) Solve the system graphically and check: y = ½ x 2 3x + 3 y = 2x
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Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts
Lesson 6 Quadratic Functions and Equations Lesson 6 Quadratic Functions and Equations We are leaving exponential functions behind and entering an entirely different world. As you work through this lesson,
Name: Date: Class Period: Algebra 2 Honors Semester 1 final Exam Review Part 2
Name: Date: Class Period: Algebra 2 Honors Semester 1 final Exam Review Part 2 Outcome 1: Absolute Value Functions 1. ( ) Domain: Range: Intercepts: End Behavior: 2. ( ) Domain: Range: Intercepts: End
Unit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all
Quadratic Functions (Section 2-1)
Quadratic Functions (Section 2-1) Section 2.1, Definition of Polynomial Function f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic
3 3.2 Investigating Quadratic Functions in Standard Form
Chapter 3 3.2 Investigating Quadratic Functions in Standard Form Focus On... identifying quadratic functions in standard form determining the vertex, domain and range, axis of symmetry, maximum or minimum
REVIEW FOR THE FIRST SEMESTER EXAM
Algebra II Honors @ Name Period Date REVIEW FOR THE FIRST SEMESTER EXAM You must NEATLY show ALL of your work ON SEPARATE PAPER in order to receive full credit! All graphs must be done on GRAPH PAPER!
9.5 Graphing Quadratics- Parabolas
Algebra G w0ka6o tkjuptnac asuoffzttwtarr]eg ]LGLXCX. q gaalxln lrxifgthftqsq Nrae_seXrcvveEdf. 9. Graphing Quadratics- Parabolas Name ID: Date Algebra Make sure each person in our group has a worksheet
Calculator Basics TI-83, TI-83 +, TI-84. Index Page
Calculator Basics TI-83, TI-83 +, TI-84 Index Page Getting Started Page 1 Graphing Page 2 Evaluating Functions page 4 Minimum and Maximum Values Page 5 Table of Values Page 6 Graphing Scatter Plots Page
WHAT ARE THE PARTS OF A QUADRATIC?
4.1 Introduction to Quadratics and their Graphs Standard Form of a Quadratic: y ax bx c or f x ax bx c. ex. y x. Every function/graph in the Quadratic family originates from the parent function: While
For every input number the output involves squaring a number.
Quadratic Functions The function For every input number the output involves squaring a number. eg. y = x, y = x + 3x + 1, y = 3(x 5), y = (x ) 1 The shape parabola (can open up or down) axis of symmetry