Quadratic Functions

Quadratic Functions Mr. Hardy Chapter 5 Algebra 2

Do Now- Homework Quiz • Chapter 4.1 (page 204) • #22, 28, 30, 34 • Chapter 4.2 (page 209) • #22, 34 • Chapter 4.3 • 20, 28 • ANSWERS ONLY • 5 MINUTES • Make 6 Coordinate Planes

ACT Practice  • In the 2 × 2 matrix below, b1 and b2 are the costs per pound of bokchoy (Chinese greens) at Market 1 and Market 2, respectively; r1 and r2 are the costs per pound of rice flour at these 2 markets, respectively. In the following matrix product, what does q represent? • F. The cost of r1 pounds of rice flour at $0.50 per pound • G. The cost of a half-pound of rice flour at Market 1 • H. The total cost of a half-pound of bokchoy and a half-pound of rice flour at Market 1 • J. The total cost of a half-pound of bokchoy and a half-pound of rice flour at Market 2 • K. The total cost of a half-pound of rice flour at Market 1 and a half-pound of rice flour at Market 2

Chapter 5 – Quadratic Functions • In this chapter, you’ll learn about • Quadratic functions, equations, and inequalities • Four ways to solve quadratic equations • How to graph quadratic functions and inequalities • Prerequisites • Solving equations • Graphing Inequalities • Graphing absolute value functions

KNOW THIS • Quadratic functions have the form • ax2 + bx + c, where a ≠ 0 • The graph of a quadratic function is U-shaped, and is called a parabola.

PARABOLAS • What are some examples of parabolic shapes?

KNOW THIS, TOO! • See the graphs of y = x2, and y = -x2 (the red and blue graphs) • The origin is the lowest point of y = x2 and the highest point of y = -x2. • The lowest or highest point of quadratic function is the vertex.

Vertex

RECALL • What is an AXIS OF SYMMETRY? • What is the SIGNIFICANCE? Explain: Why did we use it in Chapter 2.8? • The graphs of y = x2 and y = -x2 are symmetric about the y-axis. • In general, the axis of symmetry of a quadratic function is a vertical line through the vertex • JUST LIKE CHAPTER 2.8

The Graph of a Quadratic • The graph of y = ax2 + bx + c is a parabola with these characteristics • The parabola opens up if a > 0 and opens down if a < 0. The parabola is wider than the graph of y = x2 if |a| < 1 and narrower than the graph of y = x2 if |a| > 1 • The x-coordinate of the vertex is . • The axis of symmetry is x =

EXAMPLE 1- Standard Form • Graph y = 2x2 – 8x + 6 • The coefficients are: • a = 2 • b = -8 • c = 6 • First, FIND and PLOT the Vertex

EXAMPLE 1- Standard Form • The x-coordinate of the vertex is 2. • NOW, let’s find the y-coordinate of the vertex! • The y-coordinate is (Plug the x value into the quadratic function): • y = 2(2)2 – 8(2) + 6 • y = 8 – 16 + 6 • y = –2 • Therefore, the vertex is (2, -2) • What is the axis of symmetry? • x = 2

EXAMPLE 1- Standard Form • Next, find points on ONE SIDE of the axis of symmetry • Use symmetry to plot two more points - in all, you should have five points!! • Draw a parabola through the plotted points!!

y x GRAPH of y =2x2 – 8x + 6

y x Graphing a Quadratic Function STEP 1: Find the Axis of symmetry STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex.

y x y x 2 3 Graphing a Quadratic Function STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. –1 5

Y-axis y x Y-intercept of a Quadratic Function The y-intercept of a Quadratic function can Be found when x = 0. IMPORTANT The constant term is always the y- intercept

Magic Numbers!!!  • WORKS EVERY TIME (EVEN WITH FRACTIONS!) • WATCH CAREFULLY • y = -2x2 + 4x + 1 • y = -⅓x2 – 2x – 3

TRY THESE • Guided Practice, Page 253, #4, 5, 7 and 8 • Homework, Page 253, Chapter 5.1 #18-30 even

MAD MINUTE • In the 2 × 2 matrix below, b1 and b2 are the costs per pound of bokchoy (Chinese greens) at Market 1 and Market 2, respectively; r1 and r2 are the costs per pound of rice flour at these 2 markets, respectively. In the following matrix product, what does q represent? • F. The cost of r1 pounds of rice flour at $0.50 per pound • G. The cost of a half-pound of rice flour at Market 1 • H. The total cost of a half-pound of bokchoy and a half-pound of rice flour at Market 1 • J. The total cost of a half-pound of bokchoy and a half-pound of rice flour at Market 2 • K. The total cost of a half-pound of rice flour at Market 1 and a half-pound of rice flour at Market 2

Do Now • A manufacturer of lighting fixtures has daily production costs modeled by y=0.25x2 – 10x + 800 where y is the total cost in dollars and x is the number of fixtures produced. • What is the minimum daily production cost, y? • How many fixtures should be produced each day to yield a minimum cost?

Recall • Graph the Quadratic Functions • y = -x2 + 8x + 2 • y =2x2 – 12x

Mad Minute- Solving Equations • 2x – 4= 8 • 3x + 7 = -14 • x2 – 10 = 54 • ½x + 3x = 5x + 2 • 3x + 9 = 5x + 5 • 2(x – 1) = 3/5(10 + 5x) • 7x – (2x – 12) = -8

Vertex and Intercept Forms • Vertex Form of a Quadratic Function • y = a(x – h)2 + k • The vertex is (h, k) • The axis of symmetry is x = h • Intercept Form of a Quadratic Function • y = a(x – p)(x – q) • The x-intercepts are p and q • The axis of symmetry is halfway between (p, 0), and (q, 0)

Graphing a Quadratic: Vertex Form • Graph y = -½(x + 3)2 + 4 • The function is in the form a(x – h)2 + k • a = -½ • h = -3 • k = 4 • Therefore, the vertex is (-3, 4) • Since a < 0, the graph opens down • Plot two points, and use the axis of symmetry to plot to more points. Graph the parabola!

Graphing a Quadratic: Vertex Form • Plot Points • y = -½(x + 3)2 + 4 • y = -½(-1 + 3)2 + 4 • y = -½(2)2 + 4 • y = -½(4) + 4 • y = -2 +4 • y = 2 • (-1, 2) • Plot Points • y = -½(x + 3)2 + 4 • y = -½(1 + 3)2 + 4 • y = -½(4)2 + 4 • y = -½(16) + 4 • y = --8 +4 • y = -4 • (1, -4)

y x GRAPH of y = -½(x + 3)2 + 4

Try These • Graph the Quadratics in Vertex Form • y = 2(x + 1)2 + 1 • y = -3(x – 1)2 – 4

DO NOW • From 1990 to 1996, the consumption of poultry per capita is modeled by y = -0.2125t2 + 2.615t + 56.33, where t = 0 corresponds to 1990. During what year was poultry consumption per capita at it’s maximum?

DO NOW: HW QUIZ • Chapter 5.1 • Identify the Vertex ONLY • 20, 22, 24, 26, 28, 30 • What does “c” represent when graphing quadratics from standard form?

Graphing Quadratics: Intercept Form • The intercept form is: • y= a(x – p)(x – q), • The intercepts occur at (p, 0) and (q, 0) • The axis of symmetry lies halfway between these points, thus, the x coordinate of the vertex does as well • Graph y = -(x + 2)(x – 4)

y x GRAPH of y = -(x + 2)(x – 4)

Try These • Graph from Intercept Form • y = (x + 4)(x – 1) • y = -2(x – 3)(x + 1)

FOIL • FOIL, however, could come in handy. • Equations in INTERCEPT FORM, could be rewritten in STANDARD FORM using FOIL. • FIRST • OUTER • INNER • LAST

EXAMPLES • Boys • Girls

BOX METHOD Rewrite y = (2x + 5)(3x – 4) in standard form 2x + 5 3x -4 6x2 +15x -8x -20 y = 6x2 + 7x – 20

Discussion • Which ones are the same as (x + 2)2? • (x2 + 22) • (x + 2)(x + 2) • x2 + 4 • x2 + 4x + 4 • REMEMBER (x+ n)2 ≠ (x2 + n2) THIS IS REALLY REALLY REALLY BAD

Rewrite in Standard Form (Try using FOIL or the Box Method) • y = -(x + 3)(x - 4) • y = -3(x - 7)(x + 4) • y = -6(x – 2)2 – 9 • y = ½ (8x – 1)2 – 3/2

Challenge • Find a, b, and c so that the parabola whose equation is y = ax2 + bx + c has its vertex at (3, 2), and passes through the point (–1, 10).

TRY THESE • Homework • Chapter 5.1, page 253 # 18-36 even • Chapter 5.1 #38-52 all. • Try to graph them using the directions listed; however, at the end of the day, it’s up to you!

DO NOW • Rewrite in Standard Form (Try using FOIL or the Box Method) • y = -(x + 3)(x - 4) • y = -3(x - 7)(x + 4) • y = -6(x – 2)2 – 9

Challenge • Find a, b, and c so that the parabola whose equation is y = ax2 + bx + c has its vertex at (3, 2), and passes through the point (–1, 10).

MAD MINUTE • Write the functions in standard form • y = (x + 2)(x – 3) • y = (x + 1)(x + 2) • y = (x + 4)(x – 3) • y = (x – 6)(x – 8) • y = (x + 9)(x – 9)

MAD MINUTE- Finding Factors • 64 • 4 • -3 • 10 • 100 • 16 • 32 • 1 • -81 • 55 • 46 • 72 • 2 • 12

Chapter 5.2 • Solving Quadratic Equations by Factoring • Quick Notes • An expression or equation with one term is a monomial • Ex. 2x2 • Two terms – binomial • Ex. x + 5 • Three terms- trinomial • Ex. x2 + 3x -1 • Use Factoring as a way to write a trinomial as a product of binomials

Factoring • In order to factor x2 + bx + c, find integers m and n: • x2 + bx + c = (x + m)(x + n) • = x2 + (m + n)x + mn • Explain: Exactly what does this mean?

Factoring x2 + bx + c Note: • Find factors that multiply to give you the last term, and add to give you the middle term! • Both integers must be positive if b and c are positive • Both integers must be negative if c is positive and b is negative • One integers must be positive and one must be negative if cis negative

Example • Factor (x + 5) (x + 2)

Example • Factor

Quadratic Functions