Quadratic Relationships
Sarah Qarizadha
Welcome Everyone!
Learning Goals
- To be able to solve all vertex form equations
- Finding the value of 'a'
- To use quadratic equations to solve word problems in vertex form
Introduction
Definitions Of Unit #1
Parabola : This is a curve, shaped as a arch. Its distance from a fixed point is equal to its distance from a fixed line
Vertex Form : The vertex are the points (h,k) on a parabola. The vertex is when both the points meet. The x-intercept is 'k' and the y-intercept is 'h'.
AOS (Axis Of Symmetry) : The AOS is a straight vertical line that goes right down the middle of the parabola
Vertex Form
Introduction To Vertex form
- Vertex is (h,k)
- 'a' tells us if its stretched or compressed, and the direction of opening
- 'h' tells us if its going to be left or right on the graph/horizontal translation
- 'k' tells is if its a vertical translation
- Use the step pattern
- To find the y-intercept, set x=0 and then solve for the y-intercept
- Then to solve, set y=0 and solve for x or expand and simplify to get the standard form
- Then use the Quadratic Formula
Equation Question Examples
Example #1 - Determine an equation when given the vertex and determining the 'a' value:
Vertex : (-3,5)
One of the points are (3,7)
Step 1 : Plug in the vertex into the equation
y=a(x-h)^2+k
y=a(x+3)^2+5
Step 2 : Find the 'a' value. Plug in (3,7) in the equation
y=a(x+3)^2+5
7=a(3+3)^2+5
7=a(36)+5
7=36a+5
7-5=36a
2/36 = 36a/36
18=a
Step 3 : Fill in final equation
y=18(x+3)^2+5
Example #2 - Determining an equation when given the vertex and when given (x,y) :
Example #3 - Isolating For 'x' :
*We are finding the x-intercept when we are isolating for 'x'
Graphing Vertex Form
We can easily use a step pattern to graph y = 2(x-4)^2 - 6
In the step pattern, your supposed to multiply the 'a' value by 1, 3 and 5
Vertex : (4,-6)
AOS : -4
Optimal value : -6
'a' : 2
Direction Of Opening : Up, because the 'a' value is positive
By using the step pattern, we can easily figure out what parabola will be :
- Step 1 : 1x2 = 2
- Step 2 : 3x2 = 6
- Step 3 : 5x2 =10
Word Problem
Using Vertex Form
At a baseball game, a fan throws a baseball from the stadium back onto the field. The height in meters of a ball t seconds after being thrown is modeled by the function h = -4.9 (t-2)^2 + 45
a) What is the maximum height of the ball?
The Vertex : (2,45)
Maximum height of the ball is 45 meters
b) When did the maximum height occur?
The maximum height is 2 seconds
c) What is the height of the ball after 1 second?
h = -4.9 (1-2)^2 + 45
h = 40.1 meters
Therefore, the height of the ball is 40.1 meters after 1 second
d) What is the initial height of the ball?
When t=0 Solve for h
h = -4.9 (0-2)^2 + 45
=25.4
Therefore, the initial height was 25.4 meters
First and Second Differences
Mapping notation
Example - y=(x+7)^2 ~ (x-7, y)
Graphing Using Transformations
Factored Form
Learning Goals
- To be able to graph a quadratic equation in factored form
- Turning Factored Form, Quadratic Equations into Standard form by using FOIL and simplifying
A zero of a parabola is another name for the x- intercepts and in order to find the x- intercepts you must set y=0
Key points
- When the (a) value changes the zeroes do not change
- When the (a) value changes the axis of symmetry do not change
- When the (a) value changes the optimal value does change
Introduction To Factored Form
The value of r and s give you the x-intercepts
axis of symmetry, AOS: = (r+s / 2)
- Sub this x value into equation to find
- the optimal value
- to find the y-intercept, set x=0 and solve for y
- Solve using the factors
Types of Factoring:
- Greatest Common Factor
- Simple factoring (a=1)
- Complex factoring
- Special case - Difference of squares
- Special case – Perfect square
Equation Question Examples
Factoring Simple Trinomials
- Identify a,b, and c in the trinomial ax^2 + bx + c
- Write down all factor pairs of c
- Identify which factor pair from the previous step sums up to b
- Substitute factor pairs into two binomials
Factoring Complex Trinomials
Factoring by grouping
Step 1: Decide if the four terms have a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 2: Group first two terms together and the last two terms together.
Step 3: Factor out the GCF from each of the two groups.
Step 4: The one thing that the two groups have in common should be what is in parenthesis, write whats outside the brackets as a parenthesis
Step 5: Determine if the remaining factors can be factored any further.
Graphing Factored Form
Common Factoring
Step 1: Determine the GCF of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.
Step 2: Factor out or divide out the GCF from each term. You could check your answer at the point by distributing the GCF to see if you get the original question. Factoring out the GCF is the first step in many factoring problems.
Word Problem Using Factored Form
h = -5t ^2 + 25t
h = -5t (t-5)
t=0 t=5
Therefore, the rocket hit the ground at 5 seconds
Special Cases
Difference Of Squares
Equation
a^2 - b^2 = (a - b)(a + b)
or
a^2 - b^2 = (a + b)(a - b)
Step 1: Decide if theirs a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 2: So, all you need to do to factor these types of problems is to determine what numbers squares will produce the desired results.
Step 3: Determine if the remaining factors can be factored any further.
Perfect Squares
Equation
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
Step 1. Verify that the first term and the third term are both perfect squares. (This means that the coefficients are perfect squares
Step 2. Verify that the middle term is twice the product of the square roots of the first and third term.
Step. 3. Use the standard form above to write the factored form.
Factored Form - MY VIDEO
Standard Form
learning goals
- Find the number of zeros that a quadratic relationship has by calculating the discriminant.
- Solve using the quadratic formula or by factoring or by completing the square to get vertex form
Introduction to standard form
The value of a gives you the shape and direction of opening
The value of c is the y-intercept
Solve using the quadratic formula, to get the x-intercepts
MAX or MIN? Complete the square to get vertex form
Quadratic FORMULA example
- There can be two solutions, one solutions or no solution.
- There are two solutions when the discriminant is a positive.
- There is one solution when the discriminant equals to 0
- There is no solution, when the discriminant is negative
Discriminant Formula
Example Of the discriminant being used
examples of the discriminant's in graphing
Question: y=3x^2 +6x -4
- Step One: Focus on the 3x^2 +6x, and factor out the GCF (which is 3 in this case) and leave the -4 out.
y=3(x^2 +2x) -4
- Step Two: Divide the 'b' value by 2 and then square the result.
- Step Three: Add the result in the bracket and also subtract it. So the equation looks like this: y= 3(x^2 +2x +__ -__) -4
- Step Four: Take the negative one outside of the brackets, therefore it will be multiply with 3.
- Step Five: factor the trinomial in the bracket.
- *Remember Vertex Form Equation* y=a(x-h)^2+k
- Therefore, vertex is (-1, -7)
word problem using quadratic formula
a) How long does it take the rocket to fall to the ground, rounded to the nearest hundredth of a second?
b) Find the times when the toy rocket is at a height of 95.7 m above the ground. Round your answers to the nearest tenth.
c) What is the maximum height of the toy rocket? At what times does it teach this height? Round you answer to the nearest tenth.