1. Linear and Quadratic Functions
Chapter 3
Linear and Quadratic Functions

2. Section 3 Quadratic Functions and Their Properties

3. DAY 1

4. Standard form of a quadratic function: f(x) = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0. If a = 0, then ax2 doesn’t exist and you are back to a linear equation. Vertex form of a quadratic function: f(x) = a(x – h)2 + k where a, h, and k are real numbers, a ≠ 0, and (h, k) is the vertex of the graph. *The graph of a quadratic is always a parabola (U shape)*

5. Graphing using Transformations You need to use VERTEX FORM to graph using transformations! How can you tell if it is vertex or standard form? If there is a bx term or not Ex. – Which form is it in? f(x) = x2 – 2x + 2 f(x) = (x– 1)2 – 1 f(x) = x2 – 1

6. Graph using Transformations f(x) = 2x2 + 8x + 5 If not in vertex form  complete the square to put in vertex form to use transformations 1st: group x2 and x, if a ≠ 1 factor it out 2nd: complete the square (b/2)2 3rd: balance the change made

7. 4th: simplify and factor Now use transformations Basic x2  (-1, 1) (0, 0) (1, 1)

8. Properties of Quadratics To determine if it facing up or down: if a = -  down, if a = +  up All parabolas are symmetrical through the vertex Axis of symmetry x = x-value of vertex Vertex of parabola (h, k)  h = -b/2a and k = f(h) Y-intercept: find f(0) [plug in 0 for x] X-intercept: f(x) = 0 and solve for x (factor or quadratic formula)

9. SUMMARY – Steps for Graphing a Quadratic Function Option 1 Step 1: Complete the square to write in the form f(x) = a(x – h)2 + k Step 2: Graph the function in stages using transformations Option 2 Step 1: Determine whether the graph opens up or down Step 2: Determine the vertex (-b/2a, f(-b/2a)) and axis of symmetry (-b/2a) Step 3: Determine the y-intercept, f(0). Determine x-intercepts if any If b2 – 4ac > 0, then there are 2 x-intercepts If b2 – 4ac = 0 then the vertex is the x-intercept If b2 – 4ac < 0 then there are no x-intercepts Step 4: Determine an additional point by using the y-intercept and the axis of symmetry

10. Example: (#31 pg. 152) (use option 2) f(x) = -x2 – 6x Opens: Vertex: A
Example: (#31 pg. 152) (use option 2) f(x) = -x2 – 6x Opens: Vertex: A.O.S.: Y-int: Domain: X-int: Range:

11. DAY 2

12. Writing Equations of Quadratics Start with vertex form: Plug in vertex: Use another point to find a:

13. Determine if the quadratic has a minimum or maximum value and find that value. f(x) = 2x2 + 12x

14. EXIT SLIP