MATHEMATICAL METHODS UNITS 3 AND Sketching Polynomial Graphs
|
|
- Myles Morgan
- 6 years ago
- Views:
Transcription
1 Maths Methods 1 MATHEMATICAL METHODS UNITS 3 AND 4.3 Sketching Polnomial Graphs ou are required to e ale to sketch the following graphs. 1. Linear functions. Eg. = ax + These graphs when drawn will form a straight line. The gradient of this line will = a and the intercept =. Eg. = x - 1 x intercept intercept When =, = x - 1 When x =, = - 1 x = 1 = -1 1 X Note that a linear function is of one degree, and cuts the x-axis at no more than one point. -1 The gradient of a straight line joining the two points (x 1, 1 ) and (x,, ) is given m = 1 x x From this ou get the equation of the straight line joining the two points - 1 = m(x x 1 ) 1 The gradient of the slope is = tangent of the slope = x x If two straight lines are perpendicular to each other the products of their gradients is 1. The distance etween points (A and B) is given AB = The midpoint of a straight line joining (x 1, 1 ) and (x,, ) is the point 1 x1) ( 1) x 1 x 1 ( x Graphing inequations Eg. x - 1 x intercept intercept When =, = x - 1 When x =, = - 1 x = 1 = -1, X The shaded area is x - 1
2 Maths Methods 1. Quadratic Functions Eg. = ax + x + c These graphs when drawn will form a paraola. A quadratic function is a polnomial function of degree, and cuts the x-axis in no more than two points and has no more than one turning point. To sketch the graph of a quadratic function (called a paraola) use the following: If a >, the function has a minimum value If a <, the function has a maximum value The value of c gives the -intercept The equation of the axis of smmetr is x = -, this comes from writing the formula as a 4ac x = - a a Use x = -, to find the turning point a The x axis intercepts are determined solving the equation ax + x + c = The general form of the paraolas are: x x = ax = -ax a x (,a) (, -a) a x = (x a)(x ) = -(x a)(x ) Positive paraola Negative paraola A quadratic can e solved : Factorising Eg. x + 5x 1 = (x 3)(x + 4) = x = 3 or 4 Completing the square Eg x + x 4 = i.e. a = 1, =, c = -4 Add and sutract x + x +(1 1) 4 = (x + 1) 5 = to complete the square.
3 (x + 1) = 5 x + 1 = 5 x = -1 5 Using the general quadratic formula x = Eg -3x 1x 7 = x = ( 1) ( 1) ( 3) Maths Methods 1 4( 3)( 7) 6 15 x = 3 Using the discriminant of the quadratic function f(x) = ax + x + c If 4ac >, the graph of the function has two x-axis intercepts If 4ac =, the graph of the function touches the x-axis If 4ac <, the graph of the function does not intersect the x-axis a 4ac 4ac a 4ac a 4ac 4ac a a Example 1 Sketch the graph of f(x) = -3x 1x 7 using the quadratic formula to calculate the x-axis intercepts. Since c = -7 the -intercept is (, -7) Turning points coordinates: Axis of smmetr, x = - a ( 1) = - 3 = - (Use this to find the co-ordinates of the turning point) and f() = -3(-) 1(-) 7 = 5 Turning point coordinates (-, 5) - a
4 x = Maths Methods 1 To calculate the x-axis intercepts: -3x 1x 7 = a 4ac x = = ( 1) ( 1) ( 3) = 6 15 = or.71 4( 3)( 7) (-, 5) 6 4 (-3.9, ) (-.71, ) x (, -7) -8 Determining the rule for a function of a graph. All paraolas are in the form of = ax + x + c. It is possile to identif the values of a, and c from the graph. The general form of the paraolas are: (,5) x This is in the form = ax When x = =5 5 = a() a = 4 5 The rule is = 4 5 x
5 Maths Methods 1 (, 3) (-3,1) The rule is = - 9 x + 3 This is in the form = ax + c, where we know that a <. For (, 3) 3 = a() + c implies c = 3 For (-3, 1) 1 = a(-3) = 9a + 3 a = - 9 (-1, 8) 3 x This is in the form = ax + x For (3, ) = a(3) + 3 = 9a + 3 (1) For (-1, 8) 8 = a(-1) 8 = a () 8 + = a Sustitute into (1) = 9(8 + ) + 3 = s -7 = 1 = -6 a = The rule is = x 6x -1 1 x This is in the form = ax + x + c, where we know that a <. For (-1, ) = a + c (1) For (, ) = c () For (1, ) = a + + c (3) Sustitute c = in (1) and (3) = a +
6 Maths Methods 1 - = a (4) = a + + = a + (5) Sutract (5) from (4) - = - = 1 Sustitute =1 and c = into (1) = a 1 + a = -1 The quadratic rule is = -x + x + 3. Cuic functions Graphs of = ax 3 + x + cx + d A cuic function is a polnomial function of degree 3, and cuts the x-axis in no more than three points and has no more than two turning points. The shapes of cuic graphs var, ut in general the follow the following forms. x x = ax 3 = -ax 3 x - x = ax (x ) = -ax(x + ) (, ac) -a c x -a c x (, -ac) = (x + a)(x )(x c) = -(x + a)(x )(x c) In the equation = ax 3 + x + cx + d, the ax 3 term will dominate, so if a is positive, when x,
7 Maths Methods 1 Example Sketch the graph of f(x) = x 3 4x 11x + 3 Factorise the expression first using the Factor Theorem. Let x 3 4x 11x + 3 = P(1) = P(-1) = P() = = x is a factor. Dividing x 3 4x 11x + 3 x gives P(x) = (x )(x x 15) = (x )(x 5)(x + 3) x = or x 5 = or x + 3 = x = x = 5 x = -3 (, 3) -3 5 x The turning points are left undefined at this stage. = (x + 3)(x )(x 5) Determining the rule for a function of a graph. Example 3 The graph shown is that of a cuic function. Find the rule for this cuic function x This graph is in the form = k(x 4)(x 1)(x + 3). For (, 4) 4 = k(-4)(-1)3 k = 3 1 The rule is = 3 1 (x 4)(x 1)(x + 3). Example 4 The graph shown is that of a cuic function. Find the rule for this cuic function. (, 9) -3 1 x
8 Maths Methods 1 This graph is in the form = k(x 1)(x + 3). Since it onl touches the axis at this point For (, 9) 9 = k(-1)(9) k = -1 The rule is = -(x 1)(x + 3) Example 5 The graph of a cuic function passes through the points with coordinates (, 1), (1, 4), (, 17) and (-1, ). Find the rule for this cuic function. The cuic function must e of the form = ax 3 + x + cx + d For the point (, 1) 1 = d d = 1 For the point (1, 4) 4 = a + + c = a + + c (1) For the point (, 17) 17 = 8a c = 8a c 8 = 4a + + c () For the point (-1, ) = -a + - c = -a + c (3) Add (1) and (3) = 4 = Sustitute in (1) and () 1 = a + c (4) 8 = 8a + c 4 = 4a + c (5) (5) (4) 3 = 3a a = 1. From (4) c = The rule is = x 3 +x Quartic functions The general form for a quartic function is = ax 4 + x 3 + cx + dx + e The shapes of quartic graphs var, ut in general the follow the following forms. x x Flat ottom = ax 4 = -ax 4
9 Maths Methods 1 intercept is acd, ecause, here x = = a c d = acd (, acd) -a - c d x -a - c d x (, -acd) = (x + a)(x + )(x c)(x d) = -(x + a)(x )(x c)(x d) The graph will cross the -axis at the point a c d. It will ehave like = ax 4 when x is large. A repeated factor (x a) implies that the graph will touch the x-axis at a. A quartic function is of degree four and cuts the x-axis in, at most, four points and has, at most, three turning points. Example 6 Sketch the graph of the function = (x + )(x + 4)(x )(x 3) If =, then the graph crosses the x-axis at the points (-, ), (-4, ), (, ) and (3, ) If x =, the graph crosses the -axis at the point (, ) (, 48) As x gets large the graph ehaves like ax 4. (, 48) -4-3 = (x + )(x + 4)(x )(x 3) Cutting, Touching and Inflexion A polnomial function of n degrees ma: Cut the x-axis in, at most, n points Have, at most, (n 1) turning points. When factorised, if the expression is (x + a) 1, then the function cuts the x-axis at -a. if the expression is (x + a), then the function touches the x-axis at -a. +ve grad if the expression is (x + a) 3, then the point -a will e a point of inflexion. (i.e. a point of zero gradient with out a sign change of the gradient.) zero grad +ve grad
10 Maths Methods 1 Example 7 Sketch the graph of p(x) = (x 1)(x )(1 x) Use our graphics calculator to do this. Often this is a multiple choice question with a variet of graph for ou to choose from. Example 8 The graph of the function f:r R, defined f(x) = x 4 4x, has A no x-intercepts D three x-intercepts B one x-intercept E four x-intercepts C two x-intercepts The technique to solve this tpe of multiple choice question is to ignore the answers and solve the prolem ourself, and then see which answer is correct ecause it agrees with ou. x 4 4x = x (x 4) = x x(x )(x + ) the factors are x, x, x +, the intercepts are,, -. the answer is D. Example 9 The graph sketched to the right represents a polnomial function of degree 4. The rule could e A = x(x 1) ( x) B = -x(x 1)(x ) C = x(x 1) (x ) D = x (x 1)(x ) E = -x(x + 1) (x + ) It is not a good idea to tr to plot all these graphs on our calculator to tr to find the correct answer, as this will take too much time on the exam. Tr to eliminate as man possiilities as ou can. The graph touches at 1, the (x 1) factor must e squared. This leaves onl A and C. Graph C can e eliminated ecause it is a positive quartic, which means that the graph must tend to + as x +. A
11 Maths Methods 1 Example 1 Express f(x) = x 3 + x 8x 4 as the product of linear factors. ou can use factor 7 on our calculator to find the factors. The slow wa is x 3 + x 8x 4 = x (x + 1) 4(x + 1) = (x + 1)(x 4) = (x + 1)(x )(x + ) Don t forget to do this last step, the factors must e linear. Example 11 Express f(x) = 6x 4 + 5x 3 6x as the product of linear factors. ou can use factor 7 on our calculator to find the factors. The slow wa is 6x 4 + 5x 3 6x = x (6x + 5x 6) = x (3x )(x + 3) = x x (3x )(x + 3) Don t forget to do this last step, the factors must e linear. Example 1 The linear factors of x 3 4x are A x, x 4 B x, x 4, x - 4 C x, x D x, x + E x, x, x + The technique to solve this tpe of multiple choice question is to ignore the answers and solve the prolem ourself, and then see which answer is correct ecause it agrees with ou. x 3 4x = x(x 4) = x(x )(x + ) the factors are x, x, x +, the answer is E. Example 13 Sketch the graph of f(x) = x (x 9) ou should quickl sketch this on our graphics calculator, to see the answer. When ou are sketching it ou must ensure that ou clearl lael all relevant points on the graph. This means all intercepts, and make sure that ou lael the axes. ou can also think aout the answer, to confirm what the calculator is showing. This is a quartic function so it must have one of the general shapes. From the equation the graph needs to touch at the point (, ), and cross at the solutions for x 9 =. it needs to cross at x = 3. It is also a positive quartic.
12 Maths Methods 1 Example 14 The graph of the function f: R R, defined f(x) = x 4 x, has A no x-intercepts D three x-intercepts B one x-intercept E four x-intercepts C two x-intercepts To find the intercepts, ou need to factorise the expression. x 4 x = x(x 3 1) = x(x 3 1) = x(x 1)(x + x + 1) check using 4ac = 1 4 = -3 no solutions This has two real solutions, x = and x = 1, for when the function is equal to. C Example 15 Sketch the graph of the function f: R R, f(x) = x ( x)(1 + x), clearl showing all important features. Use our graphics calculator to do this, ut think aout the answer first. If ou multipl the expressions out, ou end up with a negative x 4, as x, -. The x intercepts: Let f(x) =, x =, x =, and x = -1. These are the solutions when f(x) =. It must touch at x =, since it comes from x, so this is a turning point. So the graph looks like this. Example 16 The equation which defines the graph at right could e A = (x 1)(x + 1)(x 9) B = (x 1)(x 3) C = (x 1)(x + 1)(x 3)(x + 3) D = (x 1)(x + 9) E = (x 1)(x + 1)(x + 3) Since it crosses the axis at +1 and 1, the factors are (x 1)(x + 1) = (x 1). It also touches at x = 3, must have factors (x 3) B
13 Maths Methods 1 Sketching Graphs When ou are sketching graphs, ou must include the following information on ever graph. 1. The x and axes must e laelled. The axes need to e drawn with arrow heads in the positive direction. 3. The origin must e clearl laelled. 4. If the domain is specified in the question then the graph is a line. 5. If the domain is not specified in the question then the graph should have arrowheads on either end. 6. An open circle on the end of the graph is used to show < or >. 7. A closed circle is used to show either or 8. When asked for coordinates, ou must alwas give the answer in coordinate form eg. (x, ). Don t use expressions like, crosses the x axis at All intercepts must e shown with coordinates. Example 17 If f(x) = x, sketch f(x) for the following domains. {x: x R} {x : -3 x 3} x x {x: -3 < x < 3} {x : x } x x {x: x > } x
Notes Packet on Quadratic Functions and Factoring Graphing quadratic equations in standard form, vertex form, and intercept form.
Notes Packet on Quadratic Functions and Factoring Graphing quadratic equations in standard form, vertex form, and intercept form. A. Intro to Graphs of Quadratic Equations:! = ax + bx + c A is a function
More information.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3)
Co-ordinate Geometry Co-ordinates Every point has two co-ordinates. (3, 2) x co-ordinate y co-ordinate Plot the following points on the plane..(3, 2) A (4, 1) D (2, 5) G (6, 3) B (3, 3) E ( 4, 4) H (6,
More informationUNIT P1: PURE MATHEMATICS 1 QUADRATICS
QUADRATICS Candidates should able to: carr out the process of completing the square for a quadratic polnomial, and use this form, e.g. to locate the vertex of the graph of or to sketch the graph; find
More information4.3 Quadratic functions and their properties
4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept:
More informationGRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS
GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS 1.1 DIFFERENT TYPES AND SHAPES OF GRAPHS: A graph can be drawn to represent are equation connecting two variables. There are different tpes of equations which
More informationpractice: quadratic functions [102 marks]
practice: quadratic functions [102 marks] A quadratic function, f(x) = a x 2 + bx, is represented by the mapping diagram below. 1a. Use the mapping diagram to write down two equations in terms of a and
More information1.3. Equations and Graphs of Polynomial Functions. What is the connection between the factored form of a polynomial function and its graph?
1.3 Equations and Graphs of Polnomial Functions A rollercoaster is designed so that the shape of a section of the ride can be modelled b the function f(x). 4x(x 15)(x 25)(x 45) 2 (x 6) 9, x [, 6], where
More informationLINEAR TOPICS Notes and Homework: DUE ON EXAM
NAME CLASS PERIOD LINEAR TOPICS Notes and Homework: DUE ON EXAM VOCABULARY: Make sure ou know the definitions of the terms listed below. These will be covered on the exam. Axis Scatter plot b Slope Coordinate
More informationy ax bx c y a x h 2 Math 11 Pre-Cal Quadratics Review
Math 11 Pre-Cal Quadratics Review A quadratic function can e descried as y ax x c (or equivalent forms, see elow). There are an infinite numer of solutions (x,y pairs) to a quadratic function. If we plot
More informationTest Name: Chapter 3 Review
Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x - 8y = 18 Note: Each column
More informationName Class Date. Quadratic Functions and Transformations
4-1 Reteaching Parent Quadratic Function The parent quadratic function is y = x. Sustitute 0 for x in the function to get y = 0. The vertex of the parent quadratic function is (0, 0). A few points near
More informationCHAPTER 9: Quadratic Equations and Functions
Notes # CHAPTER : Quadratic Equations and Functions -: Exploring Quadratic Graphs A. Intro to Graphs of Quadratic Equations: = ax + bx + c A is a function that can be written in the form = ax + bx + c
More informationYou should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationTHE INVERSE GRAPH. Finding the equation of the inverse. What is a function? LESSON
LESSON THE INVERSE GRAPH The reflection of a graph in the line = will be the graph of its inverse. f() f () The line = is drawn as the dotted line. Imagine folding the page along the dotted line, the two
More informationMEI Desmos Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x² 4x + 1 2. Add a line, e.g. y = x 3 3. Select the points of intersection of the line and the curve. What
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationUnderstand the Slope-Intercept Equation for a Line
Lesson Part : Introduction Understand the Slope-Intercept Equation for a Line Focus on Math Concepts CCLS 8.EE..6 How can ou show that an equation in the form 5 mx b defines a line? You have discovered
More informationMEI GeoGebra Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of
More informationLinear Topics Notes and Homework DUE ON EXAM DAY. Name: Class period:
Linear Topics Notes and Homework DUE ON EXAM DAY Name: Class period: Absolute Value Axis b Coordinate points Continuous graph Constant Correlation Dependent Variable Direct Variation Discrete graph Domain
More informationAlgebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES
UNIT LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY:
More information9.1 Linear Inequalities in Two Variables Date: 2. Decide whether to use a solid line or dotted line:
9.1 Linear Inequalities in Two Variables Date: Key Ideas: Example Solve the inequality by graphing 3y 2x 6. steps 1. Rearrange the inequality so it s in mx ± b form. Don t forget to flip the inequality
More informationMath 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
More informationObjective Mathematics
6. In angle etween the pair of tangents drawn from a 1. If straight line y = mx + c is tangential to paraola y 16( x 4), then exhaustive set of values of 'c' is given y (a) R /( 4, 4) () R /(, ) (c) R
More informationa 2 + 2a - 6 r r 2 To draw quadratic graphs, we shall be using the method we used for drawing the straight line graphs.
Chapter 12: Section 12.1 Quadratic Graphs x 2 + 2 a 2 + 2a - 6 r r 2 x 2 5x + 8 2 2 + 9 + 2 All the above equations contain a squared number. The are therefore called quadratic expressions or quadratic
More information2.1. Rectangular Coordinates and Graphs. 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions. Graphs and Functions
2 Graphs and Functions 2 Graphs and Functions 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions Sections 2.1 2.4 2008 Pearson Addison-Wesley. All rights reserved Copyright
More informationof Straight Lines 1. The straight line with gradient 3 which passes through the point,2
Learning Enhancement Team Model answers: Finding Equations of Straight Lines Finding Equations of Straight Lines stud guide The straight line with gradient 3 which passes through the point, 4 is 3 0 Because
More informationIntegrating ICT into mathematics at KS4&5
Integrating ICT into mathematics at KS4&5 Tom Button tom.button@mei.org.uk www.mei.org.uk/ict/ This session will detail the was in which ICT can currentl be used in the teaching and learning of Mathematics
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More informationMath 2 Coordinate Geometry Part 3 Inequalities & Quadratics
Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x
More informationWJEC MATHEMATICS INTERMEDIATE GRAPHS STRAIGHT LINE GRAPHS (PLOTTING)
WJEC MATHEMATICS INTERMEDIATE GRAPHS STRAIGHT LINE GRAPHS (PLOTTING) 1 Contents Some Simple Straight Lines y = mx + c Parallel Lines Perpendicular Lines Plotting Equations Shaded Regions Credits WJEC Question
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationName: Chapter 7 Review: Graphing Quadratic Functions
Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers
More informationUnit 3 Higher topic list
This is a comprehensive list of the topics to be studied for the Edexcel unit 3 modular exam. Beside the topics listed are the relevant tasks on www.mymaths.co.uk that students can use to practice. Logon
More informationA function: A mathematical relationship between two variables (x and y), where every input value (usually x) has one output value (usually y)
SESSION 9: FUNCTIONS KEY CONCEPTS: Definitions & Terminology Graphs of Functions - Straight line - Parabola - Hyperbola - Exponential Sketching graphs Finding Equations Combinations of graphs TERMINOLOGY
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationAdvanced Algebra. Equation of a Circle
Advanced Algebra Equation of a Circle Task on Entry Plotting Equations Using the table and axis below, plot the graph for - x 2 + y 2 = 25 x -5-4 -3 0 3 4 5 y 1 4 y 2-4 3 2 + y 2 = 25 9 + y 2 = 25 y 2
More informationMath 1050 Lab Activity: Graphing Transformations
Math 00 Lab Activit: Graphing Transformations Name: We'll focus on quadratic functions to eplore graphing transformations. A quadratic function is a second degree polnomial function. There are two common
More informationGraphs and transformations, Mixed Exercise 4
Graphs and transformations, Mixed Exercise 4 a y = x (x ) 0 = x (x ) So x = 0 or x = The curve crosses the x-axis at (, 0) and touches it at (0, 0). y = x x = x( x) As a = is negative, the graph has a
More informationGeometry Pre AP Graphing Linear Equations
Geometry Pre AP Graphing Linear Equations Name Date Period Find the x- and y-intercepts and slope of each equation. 1. y = -x 2. x + 3y = 6 3. x = 2 4. y = 0 5. y = 2x - 9 6. 18x 42 y = 210 Graph each
More informationTopic 2 Transformations of Functions
Week Topic Transformations of Functions Week Topic Transformations of Functions This topic can be a little trick, especiall when one problem has several transformations. We re going to work through each
More informationMathematics (www.tiwariacademy.com)
() Miscellaneous Exercise on Chapter 10 Question 1: Find the values of k for which the line is (a) Parallel to the x-axis, (b) Parallel to the y-axis, (c) Passing through the origin. Answer 1: The given
More informationQuadratic Functions In Standard Form In Factored Form In Vertex Form Transforming Graphs. Math Background
Graphing In Standard Form In Factored Form In Vertex Form Transforming Graphs Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationCore Mathematics 1 Graphs of Functions
Regent College Maths Department Core Mathematics 1 Graphs of Functions Graphs of Functions September 2011 C1 Note Graphs of functions; sketching curves defined by simple equations. Here are some curves
More informationMath-2. Lesson 3-1. Equations of Lines
Math-2 Lesson 3-1 Equations of Lines How can an equation make a line? y = x + 1 x -4-3 -2-1 0 1 2 3 Fill in the rest of the table rule x + 1 f(x) -4 + 1-3 -3 + 1-2 -2 + 1-1 -1 + 1 0 0 + 1 1 1 + 1 2 2 +
More information4. TANGENTS AND NORMALS
4. TANGENTS AND NORMALS 4. Equation of the Tangent at a Point Recall that the slope of a curve at a point is the slope of the tangent at that point. The slope of the tangent is the value of the derivative
More informationSketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
More informationSPM Add Math Form 5 Chapter 3 Integration
SPM Add Math Form Chapter Integration INDEFINITE INTEGRAL CHAPTER : INTEGRATION Integration as the reverse process of differentiation ) y if dy = x. Given that d Integral of ax n x + c = x, where c is
More informationUsing Characteristics of a Quadratic Function to Describe Its Graph. The graphs of quadratic functions can be described using key characteristics:
Chapter Summar Ke Terms standard form of a quadratic function (.1) factored form of a quadratic function (.1) verte form of a quadratic function (.1) concavit of a parabola (.1) reference points (.) transformation
More informationAlgebra I Notes Unit Six: Graphing Linear Equations and Inequalities in Two Variables, Absolute Value Functions
Sllabus Objective.4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Coordinate Plane a plane formed b two real number lines (axes) that
More informationLecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
More informationGraphing Polynomial Functions
LESSON 7 Graphing Polnomial Functions Graphs of Cubic and Quartic Functions UNDERSTAND A parent function is the most basic function of a famil of functions. It preserves the shape of the entire famil.
More informationThe Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
More information2.4 Polynomial and Rational Functions
Polnomial Functions Given a linear function f() = m + b, we can add a square term, and get a quadratic function g() = a 2 + f() = a 2 + m + b. We can continue adding terms of higher degrees, e.g. we can
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More informationMATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation
MATHS METHODS QUADRATICS REVIEW LAWS OF EXPANSION A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation a) b) c) d) e) FACTORISING Exercise 4A Q6ace,7acegi
More informationVertical Line Test a relationship is a function, if NO vertical line intersects the graph more than once
Algebra 2 Chapter 2 Domain input values, X (x, y) Range output values, Y (x, y) Function For each input, there is exactly one output Example: Vertical Line Test a relationship is a function, if NO vertical
More informationAQA GCSE Further Maths Topic Areas
AQA GCSE Further Maths Topic Areas This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need
More informationSECTION 8.2 the hyperbola Wake created from shock wave. Portion of a hyperbola
SECTION 8. the hperola 6 9 7 learning OjeCTIveS In this section, ou will: Locate a hperola s vertices and foci. Write equations of hperolas in standard form. Graph hperolas centered at the origin. Graph
More informationDeveloped in Consultation with Tennessee Educators
Developed in Consultation with Tennessee Educators Table of Contents Letter to the Student........................................ Test-Taking Checklist........................................ Tennessee
More informationTransformations. What are the roles of a, k, d, and c in polynomial functions of the form y a[k(x d)] n c, where n?
1. Transformations In the architectural design of a new hotel, a pattern is to be carved in the exterior crown moulding. What power function forms the basis of the pattern? What transformations are applied
More informationGraphs, Linear Equations, and Functions
Graphs, Linear Equations, and Functions. The Rectangular R. Coordinate Fractions Sstem bjectives. Interpret a line graph.. Plot ordered pairs.. Find ordered pairs that satisf a given equation. 4. Graph
More informationS56 (5.3) Higher Straight Line.notebook June 22, 2015
Daily Practice 5.6.2015 Q1. Simplify Q2. Evaluate L.I: Today we will be revising over our knowledge of the straight line. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line
More information3, 10,( 2, 4) Name. CP Algebra II Midterm Review Packet Unit 1: Linear Equations and Inequalities. Solve each equation. 3.
Name CP Algebra II Midterm Review Packet 018-019 Unit 1: Linear Equations and Inequalities Solve each equation. 1. x. x 4( x 5) 6x. 8x 5(x 1) 5 4. ( k ) k 4 5. x 4 x 6 6. V lhw for h 7. x y b for x z Find
More informationHFCC Math Lab Intermediate Algebra 1 SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE
HFCC Math Lab Intermediate Algebra SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE THE EQUATION OF A LINE Goal I. Use the slope-intercept form of the line to write the equation of a non-vertical line
More informationGraphing Linear Equations
Graphing Linear Equations A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. What am I learning today? How to graph a linear
More informationCHAPTER - 10 STRAIGHT LINES Slope or gradient of a line is defined as m = tan, ( 90 ), where is angle which the line makes with positive direction of x-axis measured in anticlockwise direction, 0 < 180
More informationSection 1.1 The Distance and Midpoint Formulas
Section 1.1 The Distance and Midpoint Formulas 1 y axis origin x axis 2 Plot the points: ( 3, 5), (0,7), ( 6,0), (6,4) 3 Distance Formula y x 4 Finding the Distance Between Two Points Find the distance
More informationWeek 10. Topic 1 Polynomial Functions
Week 10 Topic 1 Polnomial Functions 1 Week 10 Topic 1 Polnomial Functions Reading Polnomial functions result from adding power functions 1 together. Their graphs can be ver complicated, so the come up
More informationProperties of Quadratic functions
Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation
More informationREVIEW, pages
REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in
More informationWTS TUTORING WTS CALCULUS GRADE : 12 : PROF KWV KHANGELANI SIBIYA : WTS TUTORS CELL NO. : :
WTS TUTORING 1 WTS TUTORING WTS CALCULUS GRADE : 12 COMPILED BY : PROF KWV KHANGELANI SIBIYA : WTS TUTORS CELL NO. : 0826727928 EMAIL FACEBOOK P. : kwvsibiya@gmail.com : WTS MATHS & SCEINCE TUTORING WTS
More information5.1 Introduction to the Graphs of Polynomials
Math 3201 5.1 Introduction to the Graphs of Polynomials In Math 1201/2201, we examined three types of polynomial functions: Constant Function - horizontal line such as y = 2 Linear Function - sloped line,
More informationCoordinate geometry. distance between two points. 12a
Coordinate geometr a Distance etween two points Midpoint of a line segment c Dividing a line segment internall in the ratio a : d Dividing a line segment eternall in the ratio a : e Parallel lines F Perpendicular
More informationGraphing square root functions. What would be the base graph for the square root function? What is the table of values?
Unit 3 (Chapter 2) Radical Functions (Square Root Functions Sketch graphs of radical functions b appling translations, stretches and reflections to the graph of Analze transformations to identif the of
More information1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral
1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places. 2 2.5 3 3.5 4 When When When When When
More informationTopic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
More informationGSE Algebra 1 Name Date Block. Unit 3b Remediation Ticket
Unit 3b Remediation Ticket Question: Which function increases faster, f(x) or g(x)? f(x) = 5x + 8; two points from g(x): (-2, 4) and (3, 10) Answer: In order to compare the rate of change (roc), you must
More informationREVIEW 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!
More informationRevision Topic 11: Straight Line Graphs
Revision Topic : Straight Line Graphs The simplest way to draw a straight line graph is to produce a table of values. Example: Draw the lines y = x and y = 6 x. Table of values for y = x x y - - - - =
More informationWhat is log a a equal to?
How would you differentiate a function like y = sin ax? What is log a a equal to? How do you prove three 3-D points are collinear? What is the general equation of a straight line passing through (a,b)
More informationTIPS4RM: MHF4U: Unit 1 Polynomial Functions
TIPSRM: MHFU: Unit Polnomial Functions 008 .5.: Polnomial Concept Attainment Activit Compare and contrast the eamples and non-eamples of polnomial functions below. Through reasoning, identif attributes
More information3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System
3. The Rectangular Coordinate System Interpret a line graph. Objectives Interpret a line graph. Plot ordered pairs. 3 Find ordered pairs that satisfy a given equation. 4 Graph lines. 5 Find x- and y-intercepts.
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationgraphing_9.1.notebook March 15, 2019
1 2 3 Writing the equation of a line in slope intercept form. In order to write an equation in y = mx + b form you will need the slope "m" and the y intercept "b". We will subsitute the values for m and
More informationQuadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0
Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,
More informationCHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS
CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS Big IDEAS: 1) Writing equations of conic sections ) Graphing equations of conic sections 3) Solving quadratic systems Section: Essential Question 8-1 Apply
More informationThe Straight Line. m is undefined. Use. Show that mab
The Straight Line What is the gradient of a horizontal line? What is the equation of a horizontal line? So the equation of the x-axis is? What is the gradient of a vertical line? What is the equation of
More informationEach point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
More informationSec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.
Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical
More informationName Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =
Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations
More informationAlgebra 1 Semester 2 Final Review
Team Awesome 011 Name: Date: Period: Algebra 1 Semester Final Review 1. Given y mx b what does m represent? What does b represent?. What axis is generally used for x?. What axis is generally used for y?
More informationWJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS
Surname Other Names Centre Number 0 Candidate Number WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS A.M. MONDAY, 24 June 2013 2 1 hours 2 ADDITIONAL MATERIALS A calculator will be required for
More informationWe have already studied equations of the line. There are several forms:
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationnotes13.1inclass May 01, 2015
Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard
More informationUNIT NUMBER 5.2. GEOMETRY 2 (The straight line) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5.2 GEOMETRY 2 (The straight line) b A.J.Hobson 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2. Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises
More informationa) A(5,7) and B(3,9) b) E( 1, 4) and F( 2,8) 2) find the equation of the line, in the form y=mx+b, that goes through the points: y = mx + b
.1 medians DO IT NOW.1 Median of a Triangle 1) Determine the coordinates of the midpoint of the line segment defined by each pair of endpoints: a) A(5,7) and B(3,9) b) E( 1, 4) and F(,8) ) find the equation
More informationUnit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form
Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver
More informationName. Center axis. Introduction to Conic Sections
Name Introduction to Conic Sections Center axis This introduction to conic sections is going to focus on what they some of the skills needed to work with their equations and graphs. year, we will only
More information