warm up 1) use the graph to identify the following: a)vertex: b) zeros: c) range: d)domain: e) line...
TRANSCRIPT
Warm Up
1) Use the graph to Identify the following:
a)Vertex:b) Zeros: c) Range:d)Domain: e) Line of symmetry:f)Minimum/Maximum:
(-2,-1)
-3 and -1R: y≥-1
D: all real numbers
x=-2
Minimum=-1
LESSON 9-4
Solving quadratics by graphing
21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 23.0
California Standards
Objective
Two Zeros
Steps for solving Quadratic Equations by Graphing
One Zero No zeros
Step 1: Write the related function.
Step 2: Graph the function.
Step 3: Find the zeros of the related function
Two Roots One Root No real Roots
Finding Roots of Quadratic Polynomials
Find the roots of x2 + 4x + 3
Step 1: Write the related equation.
x2 + 4x + 3=0 y= x2 + 4x + 3
Step 2: Graph the function.(make a T table)
x y
-4
-3
-2
-1
0
3
0
-1
0
3
Step 3: Find the zeros of the related function
The roots are at -1 and -3.
SUMMARY:
What is/are the root(s)?
You try: Solve the equation by graphing the related function.
x2 + 6x + 10 = 0Step 1: Write the related function.
Step 2: Graph the function.(make a T table) y= x2 + 6x + 10
x y=x2+6x+10 (x,y)
-4
-3
-2
-1
0
(-4 )2+6(-4)+10=2
(-3 )2+6(-1 ) +10=1
(-2 )2+6(0) +10=2
(-1 )2+6(-1) +10=5
(0 )2+6(0)+10=-10
(-4,2)
(-3,1)
(-2,2)
(-1,5)
(0,10)
Step 3: Find the zeros of the related function
none
No zeros.
Solve the equation by graphing the related function.
2x2 +4x = -3 Step 1: Write the related function.
Step 2: Graph the function.(make a T table)
Step 3: Find the zeros of the related function
2x2+ 4x =-3Hint: Move all terms to one side.+3+3
2x2 +4x + 3 = 0
x y
-3
-2
-0
1
2
9
3
1
3
9
Solve the equation by graphing the related function.
2x2 – 18 = 0 Step 1: Write the related function.
Step 2: Graph the function.(make a T table) y= 2x2 – 18
x y=2x2-18 (x,y)
-2
-1
0
1
2
2(-2 )2-18=-10
2(-1 )2-18=-16
2(-0 )2-18=-18
2(-1 )2-18=-16
2(2 )2-18=-10
(-2,-10)
(-1,-16)
(0,-18)
(1,-16)
(2,-10)
Step 3: Find the zeros of the related function
-3 and 3
You try: Solve the equation by graphing the related function.
x2 – 8x – 16 = 2x2
Step 1: Write the related function.
x 2- 8x-16=2x2 Hint: Move all terms to one side.You want a positive
leading coefficient-x2+8x+16
-x2+8x+16
0= x2+ 8x +16
Step 2: Graph the function.(make a T table)
x y
-5
-4
-3
-2
-1
1
0
1
4
9
Step 3: Find the zeros of the related function
The only zero appears to be -4.
The only zero appears to be 3.
Solve the equation by graphing the related function.
-12x +18 = -2x2
Step 1: Write the related function.
Step 2: Graph the function.(make a T table)
Step 3: Find the zeros of the related function
-12x + 18x=-2x2
Hint: Move all terms to one side.You want a positive leading coefficient+2
x2
+2x2
2x2 – 12x + 18 = 0
x y
1
2
3
4
5
8
2
0
2
8
Using your graphing calculator: Find the roots of each quadratic polynomial
1) x2-x+20
2) x2-12x+35
3) x2+x-2
4) 9x2-6x+2
5) x2-4x+4
6) A frog jumps straight up from the ground. The quadratic function f(t) = –16t2 + 12t models the frog’s height above the ground after t seconds. About how long is the frog in the air?
Hint: When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t2 + 12t to find the times when the frog leaves the ground and lands.
-4 and 5
5 and 7
1 and -2
none
2
0 and 0.75
Lesson Quiz
Solve each equation by graphing the related function.
1. 3x2 – 12 = 0
2. x2 + 2x = 8
3. 3x – 5 = x2
4. 3x2 + 3 = 6x
5. A rocket is shot straight up from the ground. The quadratic function f(t) = –16t2 + 96t
models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground?
2, –2
–4, 2
ø
1
6 s