Chapter 9- Quadratic Equations
SPI3102.3.11- Analyze nonlinear graphs including quadratic and exponential functions that model contextual
problems.
SPI 3102.3.10- Find the solution of a quadratic equation and/or zeros of a quadratic function.
Section 9.1 and 9.2: Characteristics of Quadratics
Quadratic Function: any function that can be written in the standard form y = Ax2 + Bx + C
where A, B, and C are real numbers. The graph of a quadratic is a shape called a parabola.
Quadratics have 4 characteristics that need to be identified: the vertex, the maximum/minimum,
the axis of symmetry, and zeros.
Vertex (Maximum/Minimum): The highest or lowest point on a parabola.
𝐴𝑥 2 + 𝐵𝑥 + 𝐶
−𝐴𝑥 2 + 𝐵𝑥 + 𝐶
Upward
facing
Downward
facing
Minimum
Maximum
Tell whether the following will have a maximum or minimum:
1. y = 1/3 x2 + x – 3 ________________
2.
y = −3x2 + 5x
3. y = -4x2 − x + 1
4.
y = 5x2 + 2x – 6 ________________
________________
_________________
Finding a vertex:
Ex 1: Find the vertex of 𝑦 = 𝑥 2 + 6𝑥 + 9.
Step 1: Identify A, B, and C.
A: 1 B: 6 C: -4
Step 2: Find the x-value of your vertex using
−𝐵
the formula 𝑥 =
2𝐴
𝑥=
−(6)
2(1)
=
−6
2
= −3
Step 3: Plug in your x, to find the y-value of your vertex.
𝑦 = (−3)2 + 6(−3) + 9
𝑦 = 9 − 18 + 9
𝑦=0
Step 4: Write your vertex as a coordinate point and
whether it’s a max or min.
Vertex: (–3, 0); minimum
Ex 2: Find the vertex of 𝑦 = 𝑥 2 − 4𝑥 − 10
Vertex: ___________
Max or Min? __________
Axis of Symmetry: The vertical line that divides a parabola into two symmetrical halves.
Finding the Axis of Symmetry:
Ex 3: Find the axis of symmetry of the graph:
Step 1: Identify the vertex.
Vertex: (–1, 0)
Step 2: The axis of symmetry is
the x-value of the vertex.
Axis of symmetry: x = –1
Max or min: Minimum
Ex 4: Find the axis of symmetry of 𝑦 = 2𝑥 2 + 𝑥 + 3
Step 1: Find the x-value of your vertex using 𝑥
=
−𝐵
2𝐴
.
Step 2: The axis of symmetry is the x-value of the vertex.
𝑥=
−(1)
2(2)
=
−1
4
Axis of symmetry: x =
−1
4
You Try!
A) Find the axis of symmetry of the graph.
HW: TB pg. 624, #22-33
B) Find the axis of symmetry: 𝑦 = 𝑥 2 + 4𝑥 − 7
Section 9.5, 9.6, and 9.9: Solving Quadratic Equations
We are now going to learn how to solve Quadratics. To solve a Quadratic you have to find the
zeros of the function. You can do this 3 different ways: graphing, factoring, and the quadratic
formula.
Zeros: The x-value(s) that make the function equal to zero (the x-intercepts).
Finding Zeros of a Quadratic Equation: Graphing
Ex 1: Find the zeros of the graph:
By looking at a graph: You
look to see where the parabola
crosses the x-axis.
Zeros: _____________
You Try!
A) Find the zeros of the graph.
B) Find the zeros of the graph.
Finding Zeros of a Quadratic Equation: Factoring
Ex 2: Find the zeros of y = x2 - 6x + 8.
Step 1: Set the quadratic equal to zero.
0 = x2 - 6x + 8
Step 2: Factor the quadratic.
0 = (x – 4)(x – 2)
Step 3: Set each factor to zero and solve.
x-4=0
+4 +4
x=4
Zeros: x = 4 and 2
x–2=0
+2 +2
x=2
x2 + 4x = 21
Ex 3:
Ex 4:
x2 – 12x + 36 = 0
Ex 5:
-2x2 = 20x + 50
Ex D:
x2 + 4x = 5
Ex E:
3x2 – 4x + 1 = 0
YOU TRY! Solve by factoring.
x2 – 6x + 9 = 0
Ex C:
Finding Zeros of a Quadratic Equation: Quadratic Formula
Quadratic Formula:
x=
b 
b 2  4ac
2a
Solve 2x2  5x  12  0 using the
quadratic formula.
2x2  5x  12 = 0
Step 3: Simplify.
x
( 5)  ( 5) 2  4(2)( 12)
2(2)
x
Step 1: Identify a, b, and c.
a2
5
25  ( 96)
4
x
5  121
4
c  12
x
5  11
4
Step 2: Substitute into the quadratic
formula.
Step 4: Write two equations and solve.
b  5
x
( 5)  ( 5) 2  4(2)( 12)
2(2)
x
5  11
4
x4
5  11
4
or
x
or
x 
3
2
Ex 6: 6x2 + 5x - 4 = 0
Ex 7: 2x2 = 5x + 4
YOU TRY!!! Solve using the Quadratic formula.
Ex F:
-3x2 + 5x + 2 = 0
HW: TB pg. 679, #1-13, 25-27
Ex G:
3 - 5x2 = -9x