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