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, axis, and intercepts 4. Find a quadratic function given its vertex and one other point 5. Find the maximum or minimum value of a quadratic function Quadratic Function f x ax bx c where a, b, and c are real numbers and a 0 In a quadratic function there is a power of two on our independent variable and that is the highest power. Quadratic functions and applications are a large part of the mathematics of the world around us. Any general problem involving areas is a quadratic problem. The force of gravity, which basically holds the universe as we know it together, can be modeled with a quadratic function. 1
Vertex Form of Quadratic Function f x a x h k where the with a 0 vertex is hk, This is the graph of y x compressed or stretched by a translated right h units and up k units Graph of Quadratic Function In either form f x ax bx c f x ax h k if a > 0, then the parabola opens up if a < 0, then the parabola opens down Quadratic Formula The quadratic formula x b b 4ac a www.youtube.com/watch?v=b1q1ppi79ty 6
Vertex The vertex is the lowest or highest point (depending on direction) on the graph of a quadratic function b The x-value of the vertex is x a From the quadratic formula x If in form f x ax h k b b 4ac a vertex is hk, Example To find the vertex b x 8 a f x x 8x 6 y 8 6 816 13 The vertex is, Maximum The maximum or minimum of the function is always obtained at the vertex Absolute Maximum If a<0, then the parabola will open down and the vertex will be the highest point on the graph Since it is higher than all other points, not just those around it, it is an absolute maximum instead of a relative maximum Since the coordinates of the vertex are (h,k), the "absolute maximum of the function is k when x=h" 3
Minimum The maximum or minimum of the function is always obtained at the vertex Absolute Minimum If a>0, then the parabola will open up and the vertex will be the lowest point on the graph Since it is lower than all other points, not just those around it, it is an absolute minimum instead of a relative minimum Since the coordinates of the vertex are (h,k), the "absolute minimum of the function is k when x=h Stretching and Compressing In either form f x ax bx c f x ax h k a 1 0a 1 Stretches up Appears to narrow Compresses down Appears to widen Example Graph f x x 8x 5 using translations, and stretching or compressing 4 5 x x 5 x x 4 5 4 f x x x x 3 4
Graph parent function Translate left Stretch by Translate down 3 Our Result Original Function 5
Completing the Square f x ax bx c b f x ax x c a a b b b a x x c a a 4 a 4 a b b ax c a 4a Completing the Square b b 4ac f x ax a 4a Letting h and k a 4a b b 4 ac f x ax h k Summary Beginning with f x ax bx c The quadratic formula b b The vertex is, f a a b b 4ac x a The discrimant is b 4 The axis of symmetry is x ac b a The x-intercepts, if there are any, are found by solving the quadratic equation f(x) = ax + bx + c = 0 6
b 4ac > 0 If the discriminant b 4ac > 0, the graph of f(x) = ax + bx + c has two different x-intercepts. The graph will cross the x-axis at the solutions to the equation ax + bx + c = 0 Two x-intercepts b 4ac = 0 If the discriminant b 4ac = 0, the graph of f(x) = ax + bx + c has one x-intercept. The graph will touch the x-axis at the solution to the equation ax + bx + c = 0 One x-intercepts b 4ac < 0 If the discriminant b 4ac < 0, the graph of f(x) = ax + bx + c has no x-intercepts. The graph will not cross or touch the x-axis. No x-intercepts 7
Example Use the discriminant to determine the type of solutions of the equation. a.) 4x 0x + 5 = 0 a b c b 4ac = ( 0) 4(4)(5) = 400 400 = 0 There will be one real root. b.) x 3(x 8) = x 1x 3x + 4 = 0 a b c b 4ac = ( 3) 4(1)(4) = 9 96 = 87 There will be two complex roots. Parabola Axis of symmetry f(x) = ax + bx + c, a 0 a > 0 opens upward a < 0 opens downward If written in the form y k Vertex x h the vertex is at h,k Example Graph f x x 8x 6 To write in standard form, we complete the square x x f x x x 6 x 4 6 4 6 8 x The vertex is, 8
Example f x x 8x 6 To graph, we need the vertex and a couple points We found the vertex, The y-intercept is f 0 0 8 0 6 6 The x-intercepts are 0 x 3 x1 x 1,3 vertex, y-intercept 0,6 x-intercepts 3,0 1,0 By symmetry Symmetry axis x Example Suppose p ( x ) 1000 150 x is the price of a calculator as a function of the number of calculators sold and we want to maximize the revenue R( x) p( x) x 1000x150x 150x 1000x x max b 1000 a 150 70 We should sell 70 calculators 9
Find the quadratic with a vertex of Example,3 where f 1 5 Using f x a x h k we have f x a x 3 f x ax a 1 3 5 or 3 and f 1 1 3 5 a a then f x x 3 Example Find the quadratic with a vertex of,3 where f 1 5 By symmetry we have 3 1 f 3 5 vertex of,3 f 1 5 We now have three points and could use quadratic regression to find the equation End Behavior f ( x) ax bx c As x then f ( x) ax For large x values of x (positive or negative) f ( x) ax bx c ax 10
End Behavior f ( x) ax bx c As x then f ( x) ax For large x values of x (positive or negative) f ( x) ax bx c ax Example Solve x 5 x 6 0 Look for functions that be solved using quadratic methods x 5 x 6 0 x x 3 0 x 3 0 or x 0 x 3 or x x9 or x4 4,9 3 11