Section 5.1 - Properties of Quadratic Functions ♦ 217 1.6 5.1 Radical Properties Equations of Quadratic Functions Quadratic functions are not just found in a classroom. Tossing a ball in the air is a quadratic function, so is the flow rate of water in a pipe, the shape of the supporting cables of a suspension bridge and astronomical telescopes. A quadratic function is a function that can be written in the general form f ^ x h = ax 2 + bx + c where a, b and c are real numbers and a ! 0 . The graph of a quadratic function is called a parabola. s re If the equation of the parabola is written in the form y = f ^ x h = ax 2 + bx + c , the parabola opens upward if a 2 0 and downward if a 1 0 . y y nt ce C Be x x h ac The y-intercept The y-intercept is the point where the parabola intersects the y-axis. It is found by letting x = 0 in the parabola y = f ^ x h = ax 2 + bx + c . a 2 0 a10 b Pu ng hi lis There is always just one y-intercept, the point ^0, c h . The x-intercept The x-intercepts are the points where the parabola intersects the x-axis. There can be 0, 1, or 2 x-intercepts, depending on the quadratic function. m Sa x no x-intercept y y x x one x-intercept e pl y two x-intercepts THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. 218 ♦ Chapter 5 - Quadratic Functions The Vertex and Axis of Symmetry The maximum or minimum point of a parabola is called the vertex. The y-value of the vertex is the maximum or minimum value of the parabola. The vertical line that passes through the vertex of the parabola is called the axis of symmetry. If a parabola is folded along the axis of symmetry, the two sides of the graph coincide. The equation of the axis of symmetry is x = n , where n is the x-coordinate of the vertex. C y y s re Axisofsymmetry Vertex/Maximum nt ce x x Axisofsymmetry Vertex/Minimum Be Domain and Range The domain of the quadratic function y = f ^ x h = ax 2 + bx + c is all real numbers, since there is no limitation on what the x value can be. The vertex of the parabola helps find the range for the quadratic function. If the vertex is ^ x, n h , then the range is: y $ n , when the parabola opens upward. h b Pu y # n , when the parabola opens downward. y x Vertex: ^1, - 3 h Domain: All real numbers Range: y $- 3 Vertex: ^- 2, 4h Domain: All real numbers Range: y # 4 e pl x m Sa ng hi y lis ac THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. Section 5.1 - Properties of Quadratic Functions ♦ 219 Example 1 From the given graph of the parabola y = 2x 2 - 4x , determine the: a) b) c) d) e) f) g) Vertex Axis of symmetry y-intercept x-intercepts Domain of f Range of f Minimum value y x C s re ^1, - 2 h ►Solution: a) b) c) d) e) f) g) Example 2 Find the x-intercept(s), y-intercept, and axis of symmetry of the function f ^ x h = x 2 + 2x - 15 . ►Solution: x-intercept has y = 0: f ^ x h = x 2 + 2x - 15 =0 ^ x + 5 h^ x - 3 h =0 x =− 5, 3; x-intercepts ^- 5, 0 h, ^3, 0 h y-intercept has x = 0: f ^0 h = 0 2 + 2 ^0 h - 15 =− 15; y-intercept ^0, - 15 h The axis of symmetry is the line through the midpoint of x-intercepts − 5 and 3: x = - 5 + 3 =- 1 2 Example 3 Find a point on a quadratic function that has vertex ^- 1, 2 h and passes through the point ^3, - 4 h . ►Solution: Because of the symmetry of a quadratic function, when two points on the function have the same y-value, the x-values of those points must be equal distance from the x-value of the vertex. The distance from − 1 to 3 is 4 units to the right. 4 units to the left of − 1 is − 5. Therefore another point on the function is ^- 5, - 4h . nt ce x=1 ^ 0, 0 h ^0, 0 h and ^2, 0 h All real numbers y $- 2 -2 h ac Be b Pu ng hi lis m Sa e pl 4 4 THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. 220 ♦ Chapter 5 - Quadratic Functions 5.1 Exercise Set 1. Fill in the blanks. a) A function of the form y = f ^ x h = ax 2 + bx + c is called a function. b) A function of the form y = f ^ x h = ax 2 + bx + c has a graph called a . c) The lowest point on a graph that opens upward is the of the parabola. d) The vertex is the highest point on a parabola that opens . s re C e) The line that divides a parabola into two separate halves is called . f) A parabola is a polynomial of degree . g) A parabola y = f ^ x h = ax 2 + bx + c with a 2 0 , is a parabola that opens . h) A parabola y = f ^ x h = ax 2 + bx + c with a 1 0 , is a parabola that opens . Be i) The point on a parabola y = f ^ x h = ax 2 + bx + c where f ^0 h = c , is the of the parabola. ac nt ce j) The point on a parabola y = f ^ x h = ax 2 + bx + c where f ^ x h = 0 , is the of the parabola. 2. Which functions are quadratic functions? a) y = x 2 + 4 b) y = 3x - 2 c) y = x 2 + 2 x d) y = x 2 + 2 x e) y = x 2 + x f) y = x 2 + 3 3 x 3. Find the x-intercept(s) (if possible), y-intercept , and axis of symmetry of the following quadratic functions. a) y = f ^ x h = x 2 - 6x + 9 b) y = g ^ x h = x 2 - x - 2 c) y = h ^ x h = 2x 2 + x - 6 d) y = j ^ x h = 9 - x 2 e) y = k ^ x h =- x 2 + 2x + 3 f) y = l ^ x h = x 2 + 1 g) y = m ^ x h =- x 2 - 2x + 8 h) y = n ^ x h = 4 - 4x + x 2 i) y = p ^ x h = x 2 - 2 x j) y = q ^ x h =- 6 x 2 - 2 x h b Pu ng hi lis m Sa e pl THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. Section 5.1 - Properties of Quadratic Functions ♦ 221 4. Graph the given quadratic function by completing a table of values. a) f ^ x h = x 2 - 4 x b) g ^ x h =- x 2 + 4 y y x −3 −2 x C −1 0 1 1 2 2 ce −3 x x 0 3 e) k ^ x h = 1 x 2 - 6 2 ng hi lis 2 3 f) l ^ x h =- 2x 2 + 6 y x −3 −2 −2 −1 0 0 1 1 2 2 3 3 x e pl x y y m Sa y −3 −1 x −1 1 2 x −2 0 1 y b Pu −1 y −3 h −2 y ac y d) j ^ x h =- x 2 + x Be x 3 nt c) h ^ x h = x 2 - x x −1 0 s re y −3 −2 3 y THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. 222 ♦ Chapter 5 - Quadratic Functions 5. Find the following features of the given parabolas. i) Vertex ii) Axis of symmetry iii) x-intercept(s) iv) y-intercept v) Domain of f vi) Range of f vii)Maximum or minimum value C a) i) y b) i) s re y iii) ii) iii) iv) iv) v) v) vi) vi) vii) vii) ii) ce x nt c) i) ii) iii) d) i) ii) iii) iv) lis y b Pu h ac Be y v) v) vi) vi) vii) vii) x ng hi y f) i) ii) iii) ii) iii) iv) iv) v) v) vi) vi) vii) vii) x e pl m Sa e) i) x iv) y x x THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced. Section 5.1 - Properties of Quadratic Functions ♦ 223 Find a third point on the quadratic function that has the indicated vertex, and whose graph passes through the given point. a) vertex ^3, 2 h , point ^5, 7 h b) vertex ^- 1, 4h , point ^2, 5 h c) vertex ^3, 0 h , point ^7, 4h d) vertex ^2, - 5 h , point ^- 3, 2 h f) vertex ^- 1, - 4h , point ^3, - 6 h h) vertex ^- 3, 1h , point ^c, d h j) vertex ^c, d h , point ^a, b h C 6. s re e) vertex ^- 4, 2 h , point ^- 7, 8 h g) vertex ^c, d h , point ^- 3, 1h i) vertex ^a, b h , point ^c, d h 7. If the graph of a quadratic function passes through the given points, find the equation of the axis of symmetry. Does the parabola open upward or downward? a) ^0, 0 h, ^5, 0 h, ^3, 1 h b) ^- 6, 3 h, ^5, - 2 h, ^- 2, 3 h c) ^5, 2 h, ^2, 8 h, ^6, 8 h d) ^- 4, - 1h, ^- 8, - 3 h, ^0, - 1h e) ^- 7, - 5 h, ^- 3, 2 h, ^4, 2 h f) ^- 3, 4h, ^6, 6 h, ^5, 4h g) ^a, b h, ` a , c j, ^- a, bh, b 2 c 2 h) ^a, b h, ` a , c j, ^- 3a, bh, b 1 c 2 nt ce ac Be h b Pu ng hi lis m Sa e pl THEORY & PROBLEMS FOR PRE-CALCULUS 11 - Copyright © by Crescent Beach Publishing - All rights reserved. Cancopy © has ruled that this book is not covered by their licensing agreement. No part of this publication may be reproduced.