Quadratic Functions & Inequalities Chapter 5 Quadratic Functions & Inequalities

5.1 – 5.2 Graphing Quadratic Functions The graph of any Quadratic Function is a Parabola To graph a quadratic Function always find the following: y-intercept (c - write as an ordered pair) equation of the axis of symmetry x = vertex- x and y values (use x value from AOS and solve for y) roots (factor) These are the solutions to the quadratic function minimum or maximum domain and range If a is positive = opens up (minimum) – y coordinate of the vertex If a is negative = opens down (maximum) – y coordinate of the vertex

Ex: 1 Graph by using the vertex, AOS and a table f(x) = x2 + 2x - 3

Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range f(x) = -x2 + 7x – 14

Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range f(x) = 4x2 + 2x - 3

Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range x2 + 4x + 6 = f(x)

Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range 2x2 – 7x + 5 = f(x)

5.7 Analyzing graphs of Quadratic Functions Most basic quadratic function is y = x2 Axis of Symmetry is x = 0 Vertex is (0, 0) A family of graphs is a group of graphs that displays one or more similar characteristics! y = x2 is called a parent graph

Vertex Form y = a(x – h)2 + k Vertex: (h, k) Axis of symmetry: x = h a is positive: opens up, a is negative: opens down Narrower than y = x2 if |a| > 1, Wider than y = x2 if |a| < 1 h moves graph left and right - h moves right + h moves left k moves graph up or down - k moves down + k moves up

y = -6(x + 2)2 – 1 y = (x - 3)2 + 5 y = 6(x - 1)2 – 4 y = - (x + 7)2 Identify the vertex, AOS, and direction of opening. State whether it will be narrower or wider than the parent graph y = -6(x + 2)2 – 1 y = (x - 3)2 + 5 y = 6(x - 1)2 – 4 y = - (x + 7)2

Graph after identifying the vertex, AOS, and direction of opening Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = 4(x+3)2 + 1

Graph after identifying the vertex, AOS, and direction of opening Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = -(x - 5)2 – 3

Graph after identifying the vertex, AOS, and direction of opening Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = ¼ (x - 2)2 + 4

5.8 Graphing and Solving Quadratic Inequalities 1. Graph the quadratic equation as before (remember dotted or solid lines) 2. Test a point inside the parabola 3. If the point is a solution(true) then shade the area inside the parabola if it is not (false) then shade the outside of the parabola

Example 1: Graph y > x2 – 10x + 25

Example 2: Graph y < x2 - 16

Example 3: Graph y < -x2 + 5x + 6

Example 4: Graph y > x2 – 3x + 2

5.4 Complex Numbers Let’s see… Can you find the square root of a number? A. B. C. D. E. F. G.

So What’s new? For any real number x, if x2 = n, then x = ± To find the square root of negative numbers you need to use imaginary numbers. i is the imaginary unit i2 = -1 i = Square Root Property For any real number x, if x2 = n, then x = ±

What about the square root of a negative number? C. B. D. E.

Let’s Practice With i Simplify -2i (7i) (2 – 2i) + (3 + 5i) i45 i31 A. B. C. D. E.

Solve 3x2 + 48 = 0 4x2 + 100 = 0 x2 + 4= 0 A. B. C.

5.4 Day #2 More with Complex Numbers Multiply (3 + 4i) (3 – 4i) (1 – 4i) (2 + i) (1 + 3i) (7 – 5i) (2 + 6i) (5 – 3i)

*Reminder: You can’t have i in the denominator Divide 3i 5 + i 2 + 4i 2i -2i 4 - i 3 + 5i 5i 2 + i 1 - i A. D. B. E. C.

5.5 Completing the Square To complete the square for any quadratic expression of the form 𝑥 2 + 𝑏𝑥 , follow the steps listed. Step 1 Find one half of b, the coefficient of x. Step 2 Square the result in Step 1. Step 3 Add the result of Step 2 to 𝑥 2 + 𝑏𝑥 . 𝑥 2 + 𝑏𝑥 + ( 𝑏 2 ) 2 = (𝑥+ 𝑏 2 ) 2 or 𝑥 2 − 𝑏𝑥 + ( 𝑏 2 ) 2 = (𝑥− 𝑏 2 ) 2

5.5 Completing the Square Let’s try some:     Solve:    

5.6 The Quadratic Formula and the Discriminant The discriminant: the expression under the radical sign in the quadratic formula. *Determines what type and number of roots Discriminant Type and Number of Roots b2 – 4ac > 0 is a perfect square 2 rational roots b2 – 4ac > 0 is NOT a perfect square 2 irrational roots b2 – 4ac = 0 1 rational root b2 – 4ac < 0 2 complex roots

5.6 The Quadratic Formula and the Discriminant Use when you cannot factor to find the roots/solutions

Example 1: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula x2 – 3x – 40 = 0

Example 2: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula 2x2 – 8x + 11 = 0

Example 3: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula x2 + 6x – 9 = 0

TOD: Solve using the method of your choice TOD: Solve using the method of your choice! (factor or Quadratic Formula) A. 7x2 + 3 = 0 B. 2x2 – 5x + 7 = 3 C. 2x2 - 5x – 3 = 0 D. -x2 + 2x + 7 = 0