1. Quadratic Functions & Models MATH 109 - Precalculus S. Rook

2. Overview Section 2.1 in the textbook: – Properties of quadratic functions – Finding the vertex of quadratic functions – Maximum & minimum of quadratic funcitons – Applications 2

3. Properties of Quadratic Functions

4. 4 Quadratic Function Quadratic Function: a function that can be represented as f(x) = ax 2 + bx + c, where a, b, and c are constants and a ≠ 0 Key features of a quadratic function: – Vertex – Axis of symmetry Has a U-shape called a parabola – If a > 0, opens up (smile) If a < 0, opens down (frown)

5. 5 Axis of Symmetry Axis of Symmetry (x = k): a vertical line with the property that any point m units in the horizontal direction from x = k will have a companion point –m units in the horizontal direction from x = k For a quadratic function, the axis of symmetry passes through a special point called the vertex – Occurs in the middle of the parabola – Points on one side of a parabola can be reflected across the axis of symmetry to obtain a companion point on the other side

6. 6 Standard Form of a Quadratic Function Standard Form of a Quadratic Function: a quadratic function of the form f(x) = a(x – h) 2 + k, a ≠ 0 – vertex of (h, k) – axis of symmetry of x = h Obtained from f(x) = ax 2 + bx + c by completing the square Powerful form of a quadratic function because BOTH the vertex and axis of symmetry are easily accessible Can easily sketch a quadratic function in standard form using transformations (1.7)

7. Properties of Quadratic Functions (Example) Ex 1: i) Sketch the graph of the quadratic function by converting it to standard form, ii) identify the vertex, iii) identify the axis of symmetry, iv) identify the x-intercepts a)f(x) = -x 2 + 2x + 5 b)g(x) = 4x 2 – 4x + 21 7

8. Writing the Equation of a Quadratic Function in Standard Form (Example) Ex 2: Write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point a)Vertex: (3, 4); point: (1, 2) b)Vertex: (-2, 5); point: (0, 9) 8

9. Finding the Vertex of Quadratic Functions

10. 10 Vertex & Vertex Formula Depending how the parabola opens, the vertex represents the minimum or maximum point of the quadratic function – There are occasions when we need only the vertex Given f(x) = ax 2 + bx + c, the vertex is (h, k) where and – This is a much quicker way to find the vertex when we DO NOT need to sketch the parabola

11. Vertex & Vertex Formula (Example) Ex 3: Use the vertex formula to find the vertex of the quadratic function: f(x) = -2x 2 – 4x + 1 11

12. Minimum & Maximum of Quadratic Functions

13. 13 Minimum & Maximum of Quadratic Functions Depending on the value of a in f(x) = ax 2 + bx + c, the vertex is EITHER the minimum (lowest) or the maximum (highest) point of the quadratic function – If a > 0, the vertex is If a < 0, the vertex is the minimum point the maximum point – The vertex determines the range of a quadratic function

14. Minimum & Maximum of Quadratic Functions (Example) Ex 4: i) Indicate whether the quadratic function has a minimum or maximum, ii) find the minimum or maximum value, iii) state the domain and range a)f(x) = 4x 2 + 16x – 3 b)g(x) = -x 2 + 2x + 5 14

15. Applications

16. Applications for quadratic functions are fairly straightforward – Most problems ask for either the maximum or minimum value of some quadratic function Referring to the y-coordinate of the vertex 16

17. Application (Example) Ex 5: A manufacturer of lighting fixtures has daily C(x) = 800 – 10x + 0.25x 2 where C(x) is the total cost (in dollars) and x is the number of units produced. How many fixtures should be produced to yield a minimum cost AND what is that minimum cost? 17

18. Summary After studying these slides, you should be able to: – Determine which way a quadratic function will open given its equation – Write a quadratic function in standard form and then sketch it, state the vertex, and axis of symmetry – Use the vertex formula – Find the minimum or maximum value of a quadratic function and state its domain – Solve application problems Additional Practice – See the list of suggested problems for 2.1 Next lesson – Polynomial Functions of Higher Degree (Section 2.2) 18