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Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up : Factor 1. 49p 2 – 100 2. 6d 4 + 4d 3 – 6d 2 – 4d Solve for x: 3. 2x 2 + 13x + 6. Quadratic Function (y = ax 2 + bx + c). y = x 2 – x – 2. - PowerPoint PPT Presentation
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Name:Date:
Topic: Solving & Graphing Quadratic Functions/EquationsEssential Question: How can you solve quadratic equations?
Warm-Up:Factor1. 49p2 – 100
2. 6d4 + 4d3 – 6d2 – 4d
Solve for x:3. 2x2 + 13x + 6
Quadratic Function(y = ax2 + bx + c)
y = 3x2
y = x2 + 9
y = x2 – x – 2
Vocabulary:
1. Quadratic Parent Function
2. Parabola = the graph of a quadratic function is a U-shaped curved.
3. Axis of Symmetry – divide the graph into two halves
Continue
The line of symmetry ALWAYS passes through the vertex.
y
x
Vertex
Minimum
Vertex
Maximum
4. Vertex• Minimum – lowest point
of the parabola• Maximum – the highest
point of the parabola.
y = x2
a = 1, b = 0, c = 0
Minimum point (0,0)
Axis of symmetry x=0
y=x2
y = ax2 + bx + c
Find the line of symmetry of y = 3x 2 – 18x + 7
Finding the Line of SymmetryWhen a quadratic function is in standard form
The equation of the line of symmetry is
y = ax2 + bx + c,
2ba
x
For example…
Using the formula…
18
2 3x 18
6 3
Thus, the line of symmetry is x = 3.
How do I find my vertex?
Finding the VertexWhat is the vertex?
Another example:
We know the line of symmetry always goes through the vertex.Thus, the line of symmetry gives us the x – coordinate of the vertex.
To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.
STEP 1: Find the line of symmetry
STEP 2: Plug the x – value into the original equation to find the y value.
y = –2x 2 + 8x –3
8 8 22 2( 2) 4ba
x
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3
y = 5
Therefore, the vertex is (2 , 5)
STEP 1: Find the line of symmetry
Let's Graph ONE! Try …
y = 2x 2 – 4x – 1
A Quadratic Function in Standard Form
STEP 2: Find the vertex
STEP 4: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
STEP 3: Find the y-intercept.
x 2x 2 – 4x – 1 y (x, y)
Step 5: Lets Graph it!
y = 2x2 – 4x – 1
A Quadratic Function in Standard Form
y
x
What happen if we change the value of a and c ?
y=3x2
y=-3x2
y=4x2+3
y=-4x2-2
Conclusion to Quadratic Function
(y = ax2+bx+c) When a is positive,
When a is negative,
When c is positive When c is negative
the graph concaves downward.
the graph concaves upward.
the graph moves up.
the graph moves down.
Solving Quadratic Equations
Quadratic Formula
2 4
2
b b acx
a
x2 – 2x – 8 = 0
Method #1:
Hint: Quadratic equation, must equal 0
x2 – 4x = 21
Example #2 2 4
2
b b acx
a
Factoringx2 - 2x = 0
Factor in order to solve the equation.
y=x2-2x
Hints: Remember to ask yourself does the function have a GCF.Find the x intercept.
Answer: Two solutions, x=0 and x=2.
Method #2:
Page 538 (8, 10) Page 546 (43)
Page 549 (a, b, c)
Group Work:
Independent Work:
Group 1:#7, #20
Group 2: #8, #21
Group 5: #11, #24
Group 6: #12, #25
Page 544 – 545
Group 3: #9, #22
Group 4: #10, #23
Group 7: #13, #28
Group 8: #14, #30
Solve the following equations by factoring:
1. x2 = 25
2. x2 – 8 = - 7x
3. x2 – 12x = -36
4. x2 = 7x
5. m2 – 3m = 10
6. 2r – 8 = - r2
7. 4x2 = 49
8. x2 = 9x – 20
y = -x2 + 2x + 8
Find the Solutions
y=x2-4y=x2+2x-15
y=-x2+5
y=-x2-1
Find the solutions
y=x2+2x+1
y=-x2+4x-1
HLA#5:
Page 538 (7, 8)Page 544 (2, 3, 4)