S4 (4.3) Quadratic Functions.notebook February 06, 2018

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1 Daily Practice Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State the equation of the line joining (-3, -7) and (2, 8). Give your answer in the form Ax + By + C = 0 Quadratic Functions Linear functions are functions that form a straight line. E.g. y = 2x + 4 Quadratic functions are functions that form a symmetric curve called a parabola. Turning Points A Turning Point on a parabola is a point where the graph turns. y = x 2 y x Turning Point Starter Q1. Factorise x 2-5x - 24 Daily Practice Q1. Factorise x 2-81 Q2. Calculate the length of the minor arc AB A B cm Q2. Simplify Q3. Multiply out and simplify (7x - 3)(2x + 5) Q4. Simplify Q3. Write as a single fraction Q4. State the equation of the line joining (-1, 3) and (2, 5) x 0 5cm Q5. Calculate the angle at the centre given the area of the sector is 48cm 2 Q5. State the equation of the line joining (-2, 1) and (3, 5)

2 Graphs of quadratic functions in completed square form Today we will be continuing to learn about graphs of quadratic functions. Quadratic Functions To identify the turning point given the equation. Try to think about what the graph looks like. 1. The graph of the function shown is of the form y = (x - a) 2 + b, state the values of a and b. Turning Points 2. The graph of the function shown is of the form y = (x + a) 2 + b, state the values of a and b.

3 Turning Points 3. State the Turning Point and its nature of the function y = -(x + 4) 2 Daily Practice Q1. Factorise 2x 2-18 Q2. Write with a rational denominator and simplify if possible Q3.Rearrange the formula so that x is the subject Turning Points 4. State the Turning Point and its nature of the function Today we will be continuing to learn about quadratic functions. y = (x - 1) Turning Points State the turning points and their nature for the following given their equation: The equation of the axis of symmetry The axis of symmetry is a vertical line that splits a parabola in half. (a) y = x 2 (e) y = -(x - 4) (b) y = (x - 2) 2-3 (f) y = (x + 3) 2-7 (c) y = (x + 4) 2 (g) y = 4 + (x - 9) 2 (d) y = (x - 3) (h) y = -(x + 10) 2 As it is a vertical line, it has the equation x = a The line of symmetry will always go through the turning point, hence the line will always have an equation that corresponds to the x - coordinate of the T.P.

4 Daily Practice Q1. State the turning point and its nature of the function y = (x + 3) 2-1 Q2. Write x 2 + 2x - 5 in the form (x + p) 2 + q Q3. Multiply out and simplify (3x - 1)(x + 2) Today we will be continuing to learn about quadratic functions. Homework online due Q4. Factorise 2x 2-8 The equation of the axis of symmetry 1. State the equation of the axis of symmetry of the graph shown The equation of the axis of symmetry 2. State the equation of the axis of symmetry of the function y = (x - 2) Equation of the axis of symmetry State the equation of the axis of symmetry of the following given their equation: (a) y = (x - 1) 2 (e) y = -(x - 3) (b) y = (x - 5) 2-3 (f) y = (x + 4) 2-7 (c) y = (x + 7) 2 (g) y = 5 + (x - 9) 2 (d) y = (x - 10) (h) y = -(x + 12) 2 The y - intercept The y - intercept of a function is the point where it crosses the y - axis. i.e. where x = 0. To find the y - intercept, replace x with 0 and solve for y. State the y - intercept of the following 1. y = (x - 5) 2 2. y = (x + 4) y = -(x + 2) 2 + 7

5 The y - intercept State the y - intecept of the following: (a) y = (x - 1) 2 (e) y = -(x - 3) (b) y = (x - 5) 2-3 (f) y = (x + 4) 2-7 (c) y = (x + 7) 2 (g) y = 5 + (x - 9) 2 (d) y = (x - 10) (h) y = -(x + 12) 2 Daily Practice Q1. State the equation of the line joining (-3, 5) and (1, -1) Q2. Rearraneg the formula such that 'm' is the subject Q3. State the turning point, its nature, equation of axis os symmetry and the y - intercept of the function y = -(x - 2) Sketch Example: Make a sketch of the function y = (x + 1) 2-4 showing the T.P. and where the graph cuts the y - axis Today we will be continuing to learn about quadratic equations. Homework due Thursday Sketch Make a sketch of the function y = -(x - 3) showing the T.P. and where the graph cuts the y - axis Daily Practice

Quadratic Functions not in completed square form A quadratic function can also be in the form ax 2 + bx + c Today we will be learning about quadratic functions in a different form.

State where the function y = x 2 + 5x + 4 crosses the y-axis Roots of a quadratic function The are the points where a graph crosses the x - axis. At both of these points, the y - coordinate is 0.

6 Quadratic Functions not in completed square form A quadratic function can also be in the form ax 2 + bx + c Today we will be learning about quadratic functions in a different form. We can find the turning point, axis of symmetry and the y - intercept for functions in this form too. 1. State the y - intercept of the function y = x 2 + 7x State where the function y = x 2 + 5x + 4 crosses the y-axis Roots of a quadratic function The are the points where a graph crosses the x - axis. At both of these points, the y - coordinate is 0. y We can find the roots by letting y = 0. x We may need to factorise to solve. Roots of a quadratic function 2. State the roots of the function y = x 2 - x State the roots of the function y = x 2 + 4x roots Roots of a quadratic function Daily Practice State the roots of the function y = x 2 + 7x + 12

7 Roots of a quadratic function Today we will be continuing to learn about quadratic functions. Homework due tomorrow. Calendar for Improvement and Revision schoolmathematics.weebly.com Daily Practice Q1. Write x x - 2 in the form (x + a) 2 + b Q2. State the turning point, y - intercept and equation of axis of symmetry of the function y = (x - 4) Today we will be learning how to get the turning point of a quadratic y = ax 2 + bx + c Homework Due! Q3. Write with a rational denominator in its simplest form Turning Point of a quadratic function of the form ax 2 + bx + c The x - coordinate of the turning point lies half-way between the roots. To find the y - coordinate, we can substitute in our value for x. State the turning point of 1. y = x 2 + 4x Turning Point of a quadratic functionof the form ax 2 + bx + c 2. y = x 2 + 2x - 8

8 Daily Practice Sketching quadratic functions of the form ax 2 + bx + c (a) y = (x - 2)(x - 6) Sketching quadratic functions of the form ax 2 + bx + c (b) f(x) = x 2 + 2x - 3 Daily Practice Q1. Simplify Q2. Change the subject of the formula to n Q3. Solve algebraically the set of equations Using the Quadratic Formula If you cannot find the roots of a function by factorising, it usually means that the roots aren't whole numbers. e.g. x 2 + 2x + 3, we then use the Quadratic Formula: Today we will be learning about the quadratic formula 1 2 You can recognise the question because it will ask for your answer to be rounded to decimal places or surd form if Paper 1.

9 Using the Quadratic Formula Solve the following, give your answers to 1 d.p (a) x 2 + 7x - 1 = 0 Daily Practice Q1. Write as a single fraction Q2. Solve the inequality 5x - 1 < 8x - 10 Q3. Evaluate Sketching quadratic functions of the form ax 2 + bx + c (c) y = x 2 + 2x - 8 Today we will be continuing to practise using the quadratic formula. Using the Quadratic Formula (b) 7x 2-10x - 3 = 0 Using the Quadratic Formula Questions: Solve and give your answers to 1 d.p (a) 2x 2 + 4x + 1 = 0 (b) x 2 + 3x - 3 = 0 (c) 2x 2-7x + 4 = 0 (d) x 2-8x - 1 = 0 (e) x 2 + 3x - 2 = 0 (f) 3x 2 + 5x - 7 = 0

10 Daily Practice Today we will be learning about the discriminant. Issue books The Discriminant Nature of the roots: Discriminant b 2-4ac > 0 2 real distinct roots The discriminant - Gives us information about the roots of a function. b 2-4ac = 0 equal roots b 2-4ac < 0 No real roots (Imaginary roots) Nature of the roots: Discriminant 1. State the nature of the roots of the following The Discriminant Questions: State the nature of the roots for the following functions (a) x 2 + 5x + 3 = 0 (a) 3x 2-7x + 2 = 0 (b) x 2-3x + 4 = 0 (b) 2x 2-5x - 3 = 0 (c) 6x = 19x (d) 2x - 1 = x 2

11 The Discriminant Questions: Find the discriminant for each of the following and state the nature of the roots (a) x 2 + 5x + 3 = 0 Daily Practice Q1. Multiply out and simplify (2c - 3)(c + 4) Q2. Factorise x 2-144y 2 (b) 2x 2-5x - 3 = 0 Q3. State the turning point, y - intercept and equation of axis of symmetry of the function y = (x - 2) (c) 6x = 19x Q4. Write x x - 3 in completed square form (d) 2x - 1 = x 2 Functions of the form y = kx 2 If a point is on a curve, you can substitute the point into the equation of the function to find k. Today we will be learning to state equations of the graph of the form y = kx 2 1. State the value of k if the function shown is of the form y = kx 2 Functions of the form y = kx 2 Starter State the value of k if the function shown is of the form y = kx 2 Q1. Multiply out and simplify (3m 2)(2m 2 + 3m 5) Q2. Factorise 3x 2 6x Q3. State the nature of the roots of the function y = x 2 3x + 4 Q4. State the turning point and y intercept of the function y = (x 3) Q5. Solve x 2 + 7x + 12 = 0

12 Functional Notation Sometimes we use f(x) instead of y. It means the same thing. Today we will be learning about function notation. f(a) means replacing x with 'a' in your function or if we have f(4), replace x with 4. (i) Given the function f(x) = 2x + 3, find the value of f(2) (ii) Given g(x) = x 2 + 2x - 3, find g(3) 1. Evaluate Daily Practice Q1. Factorise x 2-3x + 2 Q2. Write as a single fraction 2. Solve for x Q3. State the equation of the line joining (-3, 4) and (2, 7) Q4. State the roots of the function y = x 2 + 7x + 12 Functional Notation (i) If f(x) = 3x + 2 and g(x) = 2x - 4, what is the value of x when f(x) = g(x) Today we will be continuing to practise function notation. (ii) If f(x) = x 2-4, what is the value of x when f(x) = 45? (iii) Given that f(x) = 2x - 1, what is the value of 't' when f(t) = 3?

13 Functional Notation

Finding the Equation of a Graph. I can give the equation of a curve given just the roots.

Finding the Equation of a Graph. I can give the equation of a curve given just the roots. National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today