Chapter 4 Algebra II Notes
4.1 Quadratic Functions and Transformations
Parabola: ______________________________________________________________________
Quadratic Function: y = a𝑥 2 + 𝑏𝑥 + 𝑐; 𝑎 =/= 0
Vertex Form of a Quadratic: f(x) = a(𝑥 − ℎ)2 + 𝑘
X = h is the equation for the axis of symmetry
(h,k) is the vertex of the parabola (y-value is a minimum or maximum, depends on
direction of opening)
Diagram of parent function: y = 𝑥 2 including vertex and axis of symmetry
Identifying Opening Direction – look at the a-value…positive is up and negative is down
Identifying the Vertex – look at vertex form and use h and k
Identifying the Axis of Symmetry – look at the vertex form and write is as x = h (should be xvalue of vertex)
Identifying the Domain – x-value, should always be all real numbers since there is no restriction
Identifying the Range – either all real numbers ≥ or ≤ the y-value (depends on the opening)
What are the vertex, the axis of symmetry, the maximum or minimum value, the domain, and
the range of each function?
1.) F(x) = 2(𝑥 − 4)2 + 3
2.) F(x) = -(𝑥 + 3)2 − 2.5
3.) F(x) = -2(𝑥 − 6)2
Translations of Parabolas: can use the vertical and horizontal shifts to move a parabola without
using charts; remember that we are moving from the parent function y = 𝑥 2 .
Horizontal: y = (𝑥 ± ℎ)2
Vertical: y = 𝑥 2 ± 𝑘
Stretch: number in front of a has an absolute value greater than 1
How is each graph a translation of f(x) = 𝑥 2 ?
4.) F(x) = 𝑥 2 + 4
5.) F(x) = (𝑥 + 3)2
6.) F(x) = 𝑥 2 − 2
Graphing Quadratics
1.) Identify the vertex of the equation by using the transformations we have learned and
plot it.
2.) Write the equation of the axis of symmetry and graph the line using dashes.
3.) Find and plot 2 points on one side of the axis of symmetry.
4.) Plot the corresponding points on the other side of the axis. (just count over)
Graph each function. State the vertex and the equation of the axis of symmetry.
7) y = (𝑥 − 5)2 + 2
y
Vertex:
Axis of symmetry:
x
8) f(x) = 3(𝑥 + 2)2 − 5
Vertex:
y
Axis:
x
9) f(x) = -2(𝑥 − 1)2 + 6
Vertex:
Axis:
4.2 Standard Form of Quadratics
Standard Form: f(x) = a𝑥 2 + 𝑏𝑥 + 𝑐; 𝑎 ≠ 0
Using the Standard Form to Find Vertex
1.)
2.)
3.)
4.)
List out a, b, and c.
−𝑏
Plug into x = 2𝑎 .
Use that x-value in the equation to find the y-value.
List your vertex.
Using the Standard Form to Find Axis of Symmetry:
1.) List out a, b, and c.
−𝑏
2.) Plug into x = 2𝑎 .
3.) This is your axis of symmetry.
(You can also use the x-value of the vertex in the equation x = h still)
Determining the Maximum and Minimum
1.) If a is positive the y-value in the vertex is the maximum.
2.) If a is negative the y-value in the vertex is the minimum.
Identify the vertex, axis of symmetry, maximum or minimum value, and the range of each.
Graph each.
y
1.) Y = 𝑥 2 + 2𝑥 + 1
2.) Y = -2𝑥 2 − 3𝑥 + 4
x
Converting Standard Form to Vertex Form
y
1.) List out a, b, and c.
2.) Find the vertex of the quadratic.
3.) Use “a” and (h,k) in y = a(𝑥 − ℎ)2 + 𝑘 form.
Rewrite each function in vertex form.
x
3) y = -6𝑥 2 − 12𝑥 − 1
4) y = 4𝑥 2 − 16𝑥 − 5
4.3 Modeling With Quadratic Functions
When you are given the vertex and a point, you can write in vertex from.
What if you’re not given the vertex? You must have 3 points to use in standard form.
What is the standard form of a parabola?
Steps for Using 3 Points
1.) Plug each point in for x and y and make 3 equations.
2.) Use the techniques learned to solve systems with 3 variables.
3.) Find a, b, and c and plug into the standard form.
Find an equation in standard form of the parabola passing through the following.
1.) (0,-4), (1,1), (2,8)
2.) (0,0), (-1,-2), (1,6)
3.) (1,0), (2,-3), (3,-10)
4.4 Factoring Quadratic Expressions
What multiplication method gets a quadratic expression?
What does it mean to factor a number?
What does it mean to factor a quadratic?
Factoring Quadratic Expressions
A𝑥 2 + 𝑏𝑥 + 𝑐 𝑤ℎ𝑒𝑛 𝑎 = 1
A𝑥 2 + 𝑏𝑥 + 𝑐 𝑤ℎ𝑒𝑛 𝑎 ≠ 1
Identify c
Divide any common factors
out first
Multiply a to the end with c,
should be a new quadratic
Find factors of the new end
term that add to b
Find the factors of c that add
to b (watch signs!)
Write answer in form of
binomials (x ± f1)(x ± f2)
Write in factored form
Divide each factor by a from
beginning (any denominator
will go in front of the x)
Write each expression in factored form.
1.) 𝑥 2 − 𝑥 − 20
2.) 25𝑥 2 + 30𝑥 + 9
Missing eithr b or c (common
factors)
Write each term in factored
form
Circle common factors in
each term
Write common factors down,
then write other terms in
parentheses
Simplify
3.) 4𝑥 2 − 2𝑥
Factoring Special Quadratics (Perfect Square Trinomials and Difference of Squares)
𝑎2 − 𝑏 2
Make certain both are
squares
Make certain it’s subtraction
Take the square root of each
number
𝑎2 + 2𝑎𝑏 + 𝑏 2
Make sure ends are perfect
𝑎2 − 2𝑎𝑏 + 𝑏 2
Make sure ends are perfect
Take square root of each end
Multiply: 2*a*b; is it the
same as the middle term?
Take square root of each end
Multiply: 2*a*b; is it the
same as the middle term?
(disregard sign)
Factored form is (𝑎 − 𝑏)2
Factored form is (a + b)(a – b) Factored form is (𝑎 + 𝑏)2
Write each expression in factored form.
4.) 9𝑥 2 − 1
5.) 9𝑥 2 + 48𝑥 + 64
6.) 4𝑧 2 − 20𝑧 + 25
4.5 Solving Quadratic Equations by Factoring
Zero Product Property: If a*b = 0, then a = 0 or b = 0.
In other words: once you have factored form equal to zero, set each quantify to 0 and solve.
Solving Quadratics by Factoring
1.) Make certain equation is in standard form and equal to 0.
2.) Factor using one of the techniques discussed previously.
3.) Set your factors each to 0 and solve.
Solve the following by factoring. Make certain you are showing your work.
1.) 𝑥 2 − 8𝑥 − 20 = 0
2.) 𝑥 2 − 49 = 0
3.) 5𝑥 2 + 30𝑥 + 14 = 2 − 2𝑥
Writing Equations Given Roots
Roots are the values that make an expression true. We found roots in the previous examples.
Things to remember:
If a root was positive, the factored form was (x – h).
If a root was negative, the factored form was (x + h).
Why is the sign inside opposite?
Steps for Writing an Equation with Roots Given:
1.) Take the first root and write the expression in factored form (x – h) or (x + h).
2.) Repeat step one, but with the 2nd root.
3.) Multiply the two quantities using FOIL.
Write the equation of the polynomial that has the given roots.
4.) 2 and -4
5.)
1
2
𝑎𝑛𝑑 8