Understanding Quadratic Functions: Graphing, Vertex, and Properties

1. Bellwork • Factor • 2x2+5x+3 • x2+15x-16 • Complete the statement • A function’s domain is the collection of the _________. • A function’s range is the collection of the __________.

2. Bellwork Solution • Factor • 2x2+5x+3 • x2+15x-16

3. Bellwork Solution • Factor • x2+15x-16

4. Bellwork Solution • Complete the statement • A function’s domain is the collection of the _________. • A function’s range is the collection of the __________. Inputs or x’s Outputs or y’s

5. Graph y=ax2+c Section 10.1

6. The Concept • In chapter 9 we worked with second order polynomials. • In chapter 10 we actually graph these functions and use the for analysis

7. Definitions • A second order polynomial’s graph is what is called a parabola • Thus these functions are sometimes called parabolic functions • Another term that is used is a quadratic function. • They look like this Interesting fact: Satellite dishes are parabolic in form because of the special properties attributed to these “conic sections”

8. Definitions • Much like linear functions, these functions come with their own nomenclature • Two important terms • Vertex: Highest or lowest point of a quadratic • Axis of symmetry: Imaginary line that divides the parabola into two mirrored halves Axis of symmetry Vertex

9. More terminology • These terms are important because they’re used to describe different parabolas, much like slope was used for lines • As well, they are used to describe changes made to the parent function • A parent function is a standard graph for a basic function • This graph is the parent function for a quadratic

10. Y X Graphing • At this point, graphing these function is best done via T-table x y=x2 1 1 2 4 3 9 -1 1 -2 4 -3 9

11. Y X Graphing • We can also plot several iterations to see the effect of a scalar (or leading coefficient) attached to the term • This scalar makes the equation y=ax2 x y=x2 y=2x2 y=1/2x2 1 1 2 .5 2 4 8 2 3 9 18 4.5 -1 1 2 .5 -2 4 8 2 -3 9 18 4.5

12. Y X Graphing • These graphs lead us to understand a fundamental of graphing • If a>1, the graph stretches • If a<1, the graph flattens

13. Y X Graphing • Let’s look at what happens when a<0 x y=x2 y=-x2 1 1 -1 2 4 -4 3 9 -9 -1 1 -1 -2 4 -4 -3 9 -9 Therefore we see that if a<0, the graph is mirrored over the x-axis

14. Fundamental Rules • At this point we see some fundamental rules of quadratics • If the leading coefficient is positive (a>0) • Concave up (cupped upwards) • If the leading coefficient is negative (a<0) • Concave down (cupped downwards)

15. Y X Graphing • Let’s look at one last thing • What do you think happens when we add a constant? x y=x2 y=x2+2 y=x2-3 1 1 3 -2 2 4 6 1 3 9 11 6 -1 1 3 -2 -2 4 6 1 -3 9 11 6 Therefore we see that the constant dictates the height of the function on the y-axis

16. Fundamental Rules • At this point we see some fundamental rules of quadratics • If the leading coefficient is positive (a>0) • Concave up (cupped upwards) • If the leading coefficient is negative (a<0) • Concave down (cupped downwards) • A constant added indicates the y-coordinate of the vertex

17. Homework • 10.1 • 1-5, 7-23 odd, 33-36

18. Y X Example • Graph

19. Most Important Points • A second order polynomial can be called a quadratic function • It’s graph is called a parabola • Parabola’s have a vertex and axis of symmetry • Leading coefficients either flatten or stretch graphs • Negative leading coefficients cause the graph to be mirrored across the x-axis • A constant indicates the vertex’s y-coordinate