Alg 2 H
Chapter 4 Notes
Graph, label vertex, and axis of symmetry..
Step 1. for graphing is to find the parabolas vertex. NEED „x‟ and y‟
4.1 Graph Quadratic Functions in Standard form …y = -3x2 + 18x – 5 p241-#36
b
(To find “x” you should use x = - ) b=18, a= (-3)
2a
4.2 Graph Quadratic Functions in Intercept form…y = 4(x – 7) (x + 2) p249-#21
pq
( To find “x” you should use x = ) p=7, q= (-2)
2
Graph Quadratic Functions in Vertex form …y = 3(x – 3)2 – 4 p 249 -#33 (h, k)
No “x” formula needed. Your “x‟ value is derived right from the function ( 3, -4)
Step 2. Create a table.
Substitute different “x” values into the function that are close to the “x” value in your
vertex. The table will give you new points to plot for completing the graph.
(h, k)
2
Ex. y = 3(x – 3) – 4 Vertex = (3, -4), so I can create a table D;{ 3, 2, 1} substituting 2
and 1 into the function in vertex form and then solve for “y”. You should find two more
points. Plot those and then graph the whole image based on axis of symmetry.
Solve by Factoring.
4.3 Solve x2 + bx + c = 0 by Factoring
Ex. x2 + 2x – 35 (x – 7)(x +5)
Factoring and zero property if directions say solve.
a. Difference of two squares…. x2 – 49 = (x – 7) (x + 7)
b. Perfect square trinomials…..s2 – 26s + 169 = (s – 13) (s - 13)
c. Zero product property…x2 + 2x +1 = (x + 1)(x + 1) now set both equal to zero.
x+1=0 x+1=0
x = -1 x = -1
4.4 Solve ax2 + bx + c = 0 by Factoring
Perform algebra steps on equation before factoring if needed.
Solve the equation Ex. 14s2 – 21s = 0
7(2s2 – 3s) = 0 GCF
7s( 2s – 3 ) = 0 Factored out the s
7s = 0 2s – 3 = 0 set both equal to zero
3
s=0 s=
2
4.5 Solve Quadratic equations by finding Square Roots
Ex. 2(x + 2)2 – 5 = 8
4.6 Perform operations with Complex Numbers (Remember your “fab” four)
Complex Conjugates:
Plotting in complex plane (x, y) = (r, i)
Absolute values |z| = a2 + b2
Ex. |10 + 7i| = 102 + 72 =
Write the quotient in standard form…we need to use conjugates:
7 4i
Ex.
2 3i
Write in standard form.
Ex -8 – (3 + 2i) – (9 – 4i) change the operations and combine like terms{-20 + 2i}
-8 + (-3) + (-2i) + (-9) + 4i =
Ex. (3 + 2i) + (5 – i) + 6i drop the ( ) then combine like terms {8 + 7i}
Ex. (-1 – 5i) (-1 + 5i) FOIL
Ex. 6i(3 + 2i) Distributive property
4.7 Complete the Square ( b )2 Look for algebra steps first.
2
Ex. 4x2 – 40x -12 = 0
Complete the Square
Ex. 2k2 + 16k = -12
4.8 Use the Quadratic Formula and the Discriminant
Quadratic Formula
Discriminant
a. b2 – 4ac > 0 two real solutions
b. b2 – 4ac = 0 one real solution
c. b2 – 4ac 0 Change to an equation y = 0
2
x + 6x + 3 = 0 Then use quadratic formula to find two x values
If the inequality is 0 then the solution is an OR statement
Solve a quadratic inequality algebraically
Ex -3x2 + 10x > -2 Change to an equation (you want y = 0)
-3x2 + 10x = -2 After adding two to both sides then use quadratic formula
The two values (critical points) place on a number line.
Test the values from each interval and see where the
inequality is true {-.19< x < 3.52}
4.10 Write Quadratic Functions and Models
Extra examples
1.
i14 = Divide exponent by 4 (14/4 = 3 remainder 2)
The remainder becomes the new exponent
i2 = -1
2. x2 + 4 = 2 Remove the fraction.
8 Multiply everything by 8