Quadratic Functions and Models

1. Quadratic Functions and Models Section 3.1

2. Objectives • Learn basic concepts about quadratic functions and their graphs • Apply the vertex formula • Sketch a quadratic function • Solve applications and model data • Use quadratic regression to model data.

3. Basics • The quadratic is a nonlinear function. • Its standard form is f(x) = ax2 + bx +c • The vertex form is f(x) = a(x – h)2 + kwhere (h,k) is the vertex

4. Identify the function • Y = 5x – 1 • Y = 2x2 + 1 • Y = (3x2 + 1)2 • y = 1/(x2 – 4)

5. Graph of a Quadratic Function

6. Shape of Graph • Called a parabola • If a is +, then graph opens up • If a is -, then graph opens down • The larger |a| is, the skinnier the graph. • The smaller |a| is, the fatter the graph.

7. Vertex A

8. Axis of Symmetry

9. How to Find the Vertex • Y = -3(x – 1)2 + 2 • The x-coordinate is the 1 and the y-coordinate is the 2, so (1,2) • The negative is part of the vertex form and not part of the vertex. • Y = 5(x + 2)2 – 5 • Rewrite in vertex form. Y = 5(x – (- 2))2 – 5 • So the vertex is (-2, -5).

10. How to Find the Axis of Symmetry • The axis of symmetry is a vertical line through the vertex. • Since it is a line, it has an equation. • Since it is vertical, it is always in the form of x = h (the x-value of the vertex).

11. Example • The sign of the leading coefficient (a) is ? • What is the vertex? • What is the equation of the axis of symmetry?

12. How to Find the Vertex • When quadratic function is in standard form, use the vertex formula to find the vertex. • Vertex formula • X-coordinate found by –b/2a, where a is the leading coefficient (x2 term) and b is the coefficient of the x term. • Y-coordinate found by plugging above answer in standard form and solving for y.

13. How to Find the Vertex • Y = 3x2 – 4x + 1 • a = 3 and b = -4 and c = 1 • X-coordinate = -b/2a = -(-4)/2(3) = 4/6 =2/3 • Y-coordinate = 3(2/3)2 – 4(2/3) + 1 = -1/3 • So the vertex is (2/3, -1/3) • Y-coordinate can also be found by(4ac – b2)/(4a).

14. How to Write in Vertex Form • Identify the leading coefficient, a. • Find the vertex. • Put in the vertex form. • Example • Y = x2 – 7x + 5 • A = 1, Vertex = (3.5, -7.25) • So the vertex form of the function isy = 1(x – 3.5)2 – 7.25

15. How to Get Formula From Graph • Identify vertex (-2,-2) and plug into vertex form. • Pick another obvious point, and plug into x and y in vertex form. • Solve for a. • Then put vertex and a back in vertex form.

16. How to Get Formula From Graph • Vertex (-2,-2) • (2, 8) • 8= a(2 – (-2))2 – 2 • 8 = a(4)2 – 2 • 8= 16a – 2 • 10 = 16a • 10/16 = a = 5/8 • y= 5/8(x + 2)2 – 2

17. ORRRRRR • You can plug the vertex and two other points into L1 and L2 in your calculator and do a QuadReg. This will give you a. • Suppose the vertex is (-1, 3) and the other points on the graph are (-2, 1) and (0,1). • Stat/edit, under L1 put -1 and -2 and 0. Under L2 put 3 and 1 and 1. • Stat/calc, #5 QuadReg, get the a value.

18. Example • Sign of the leading coefficient? • Vertex? • Axis of symmetry? • Increasing/ decreasing intervals?

19. #83 page 185 • A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enclosed by the fence? • Perimeter = 2L + 2W • 1000 = 2L + 2W • 500 = L + W • 500 – W = L • Area = LW • A = (500 – W)W • A= 500W – W2

20. #83 page 185 • Area is a quadratic function with x values of width of the enclosure and y values of the area corresponding to that width. The graph of the function opens downward, so it has a maximum value…at the vertex. • Find the vertex of the parabola, the x value is the width and the y value is the area. • Plug x into 500 = L + W to get the length • 250 feet x 250 feet