Quadratic Relationship

By, Drasti Patel

Why are Quadratic Relations important?

Quadratics has many practical applications in the real world. You may think that it's very insignificant, but many important and high-paying jobs like aerospace engineer, industrial engineers, etc. require an in-depth knowledge of quadratic relationships. Without quadratics it would be impossible to build safe roller coasters at Canada's Wonderland (amusement park), we wouldn't even have all of the huge, strong bridges and buildings without this important unit of mathematics.
Parabolas in real life

What are quadratic Relationships?

Quadratics in math are equations or problems that form u shaped graphs, also known as parabolas. The name Quadratics comes from the word "quadratum" which mean squared in latin. An example of a quadratic equation is: y=2(x-6)^2+2. A relation can be written in 3 forms: vertex form, factored form and standard form.

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This shows you the 3 different way to write a quadratic function. Each of the equations gives you different information about the parabola. Vertex form gives you the vertex, Standard form gives you the y-intercept and the factored form gives you the x-intercepts of the graph.

Do all graphs have a linear relation?

No, not all graphs have a linear relation because not all graphs form straight lines. For example, a quadratic relation forms a curved graph, which is called parabolas. A linear relationship is modelled by the function y=mx+b and, a quadratic relation’s base graph is modelled as y=x^2.

How do I know if a relation is linear or Quadratic?

You can make a table of values to identify if a relationship is linear or non-linear. If the first differences from the table are constant, then the relationship will be linear and will form a straight line. If the first differences aren’t constant, but the second differences are, then the relationship would be quadratic, which would form a u-shaped graph called a parabola. If neither of the first or second differences is constant then the relationship would be identified as “neither”.
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How to Tell if a Table is Linear, Quadratic, or Exponential

Parent Function

To graph the parent function Y=X^2 we first make to make a table of values. Fill in the x values with integers , then square the x values to find the y- values,we do this because the parent function is y=x^2. The table of values gives you the coordinates that help you graph the basic quadratic function. You can observe the parent function parabola below.
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Quadratics Terminology

Vertex

- The vertex of a parabola is the maximum or minimum point on the graph.

- The vertex is also the point where the parabola meets the axis of symmetry and the optimal value

- In the vertex form of a quadratic function (y=a(x-h)^2+k) the h and k, (h,k) is the vertex of the parabola. This is because the axis of symmetry is the h value of the function and the optimal value is the k value of the function.


Optimal Value

- The optimal value of a quadratic equation is the y-coordinate of the vertex.

- The optimal value is also known as k-value in the vertex form of the quadratic function.

- The optimal value is the highest or lowest point on a parabola.

- The optimal value is written as y=k because it’s the y-coordinate of the vertex


Maximum Value

- The maximum value is the highest point on the parabola.

- When the parabola has a maximum value the direction of opening is down.


Minimum Value

- The minimum value is the lowest point on the parabola.

- When the parabola has a minimum value the direction of opening is up.

Y- intercept

- The y-intercept is the point where the parabola crosses the y- axis.


X-intercept

- The x-intercept is where the parabola crosses the x- axis.


Axis of Symmetry

- The axis of symmetry passes through the vertex and divides the parabola into two equal halves.

- The axis of symmetry is the x-coordinate of the vertex, which is also the h value of a quadratic function in vertex form, therefore, the axis of symmetry is written as x=h.


Zeroes (roots)

- The zeroes of a parabola are also known as x- intercepts and, roots.

- Zeroes are spotted when the parabola crosses the x- axis of the graph.

- A parabola has a possibility of containing 0, 1 or a maximum of 2 zeroes.


The picture below shows all the 3 possibilities of the zeroes

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There're few worksheets attached below for practice.

Unit 1: Graphing in vertex form

Summary Of Graphing in Vertex Form

- Vertex form is written as y=a(x-h)^2+K

- The"" tells you the direction of if the opening is up or down

- The "h-value" tells horizontal translation. Flip the sign of the "h- value" and now the x-coordinate of the vertex

- The "" is the vertical translation of the graph

- Can use Step pattern to graph, multiply "a-value" by 1,3,5 to find the steps

- Can also use mapping notation to graph a quadratic function. Equation of mapping notation (x+h, ay+k)

- The vertex is known as (h,k)

- The axis of symmetry is x=h

-The optimal value is y=k

- Where parabola crosses the x-axis, those points are known as x-intercepts

- To find y-intercept x=0 and solve for y

Learning Goals for Graphing in Vertex Form:

1. Can identify all parts of a parabola, (i.e. Vertex, zeroes, y-intercept, axis of symmetry, etc.)

2. Can identify all important features of a quadratic function, (i.e. vertex, y-intercept, etc.)

3. Can identify linear and quadratic relations by using finite differences.

4. Can use a table of values to create a graph.

5. Can identify all transformations of a quadratic relation correctly, with explanations.

6. Can graph a quadratic equation using mapping notation.

7. Can graph a quadratic equation using step pattern.

How is a quadratic equation written?

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This is a representation of the quadratic relation in vertex form. The a- value determines if the parabola is opening up or down and if the parabola is being stretched or compressed. The h is the x- coordinate of vertex its makes the parabola move left or right. The k- value is the y- coordinate of the vertex, it tells us if the parabola shifts up or down. The vertex of a quadratic relation in vertex form is written as (h,k). It's very important to understand these key points!
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Transforming Parabolas

Vertical Stretch and Vertical Compression (affected by a- value)

- The a- value of a quadratic function tells us if the parabola is vertically stretched or vertically compressed. Many students may students describe the vertical stretch and vertical compression with the terms “wide” and “narrow”. If the a- value is more than 1 the parabola is will vertically stretched. If the a- value is less than 1 the parabola will be vertically compressed.


Direction of Opening (affected by a- value)

- The a- value tells us the direction of the opening of the parabola. If the a- value is negative the parabola is opening down, and if the a- value is positive the parabola will open up.

Ex. if the parabola is 3 the parabola

Horizontal Shift (affected by h- value)

- The h- value tells us if the parabola shifts to the right or the left. If the h- value is -3 you would first flip the sign to +3 and move the parabola 3 spaces (units) to the right. You always flip the sign of the h- value from the original equation, before you graph the parabola. Hint: If h is negative you move to the right. If h is positive you move to the left.


Vertical Shift (affected by k- value)

- The k- value moves the parabola up or down. If the k- value is negative the parabola shifts down. If the k- value is positive the parabola shifts up.

- The k- value is the optimal value of the graph.

4.1 Quadratic Functions - Transformations

The worksheet below is provided for practice.

Graphing Quadratics in vertex form

Step Pattern:

Step pattern is a method used to graph a quadratic relation. This method consists of first plotting the vertex because this determines where the start of the step pattern begins. Then we determining the step pattern, plotting the step points on the graph, then reflect each point across the axis of symmetry, and finally drawing a curved line through the plotted points to make the parabola. Don't worry, it's very easy!

First, let's start off with graphing the parent function of a quadratic relation: y=x^2.

To graph y= x^2 we will need to make a table of values, to find the coordinates of the parabola. After we get the coordinates from the table of values, we plot them on the graph.


Question: If the value of x was 7, what would be the value of the y- value?

Answer: If x=7 then y=49 because y=x^2 so y=7^2 which is 49.

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In the basic parabola above, you can see that the points go up by 1, 3, 5, etc. which is the pattern. First, we go over 1 and up 1, then over 1 and up 3 and over 1 and up 5 because those are the basic steps of the basic parabola.


After you have plotted one side of the parabola by going either right or left, you use the axis of symmetry and reflect those points to the opposite side to form the parabola.

Let's graph y=2(x-3)^2+2 using Step Pattern
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For this quadratic function, you can also make a table of values to graph the quadratic equation.

But, you can just use the step pattern instead, it faster and easier.

To find the step pattern of a function you have to multiply the a- value of the function by 1, 3, and 5.

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First Step: We need to find and plot the vertex on the graph. The vertex is (3,2). Also, identify the direction of opening, in this case, the parabola will be opening up because the a- value is positive.

Second Step is to determine the steps by multiplying 1,3 and 5 by the a- value. The step pattern for this function is 2, 6, and 10.

Third Step: Plot the steps. Start at the vertex (3,2), and then follow step pattern so, go over 1 and up 2 for the first point, go over 1 and up 6 for the second point and over 1 and up 10 for the third point.

Fourth Step: After you plotted the points for one side of the parabola, you reflect the points to the other side of the axis of symmetry.

Fifth Step: Then just draw a smooth curved line that goes through all of the points to form the parabola.

Sixth Step: Label the parabola.

Graphing a Quadratic Function using Step Pattern

Mapping Notations

Mapping Notations is an algebraic method used to accurately graph any quadratic relation. Before we can use mapping notation we need to know the “key” points of the basic quadratic relation.

The mapping formula is x+h, ay+k
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Let’s describe the transformations for y = 2(x-3)^2 - 4

a=2, vertical stretch is 2 and direction of opening is up

h=3, horizontal translation is 3 units right

k = -4, vertical translation is 4 units down


Now we can see how the various transformations above affects the coordinates of the key points of the graph of y=x^2.

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What happens to the "key" points of y=x^2 in order to become y=2(x-3)^2-4

(x,y) ------> (x+3), (2y-4) because the mapping formula is x+h, ay+k .

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Now that we have made table of values for the key points for base function (x,y) and the mapping formula (x+3, 2y-4) so we can start sketching the new graph.

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This is the graph of y=2(x-3)^2-4. Therefore, the possibilities x- values are all real numbers, and the y- value if y≤ 4 (less than or equal to 4)
Graphing Quadratic Functions Using Transformations

Word Problems in vertex form

How to determine the y- intercept?


Equation: y=-4(x-10)^2+144


To determine the y-intercept sub 0 into the x- value and solve for y.

Y-intercept: set x=0


y=-4(x-10)^2+144

y=-4(0-10)^2+144

y= -4(-10)^2+144

y=-4(100)+144

y=-400+144

y=-256

Therefore, the y-intercept would be (0, -256).

How to determine the x-intercepts?


To determine the x-intercepts, sub in 0 for y then solve for x.


Equation: y=-4(x-10)^2+144


X-intercept: set y=0


y=-4(x-10)^2+144

0-144=-4(x-10)^2+144-144

-144=-4(x-10)^2

-4 --------- -4

+ √36= √(x-10)^2

+ 6+10= x-10+10


1. +6+10=x 2. -6+10=x

+16=x 4= x


Therefore, the x-intercepts of this quadratic equation are (16,0) and (4,0).

Let's do an example of a word problem.


Question:

A baseball is hit by a baseball bat. The height of the ball is modelled by the function

h = -5(t – 1)^2 + 7, where t is the time in seconds and h is the height in metres.


Vertex: (1,7)

  1. What is the initial height of the baseball?
  2. What is the maximum height?
  3. What is the height of the ball after 2 seconds?
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Let's do another example.
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Let's do one last example.


Question: The height of a flare is a function of the elapsed time since it was fired. An expression for its height is h= -5(t-10)^2+500,


a) What is the maximum height of the flare?

b) At what time does the flare reach the maximum height?

c) Determine the height of the flare at 5 seconds.

d) Determine the time when the height of the flare is 200 metres above the ground.

e) Determine the time when the flare hits the ground.

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Go to the link below for extra word problems and a quick quiz.

Quadratics in Factored Form

Summary of Quadratics in Factored Form

- The Factored form is written as y=a(x-r)(x-s)

- a- value gives you the shape of the graph and the direction of opening of the parabola

- The values of "r" and "s" give you the x-intercepts (zeros) of the parabola

- Formula of axis of symmetry is x=r+s/2

- To find the x-intercept replace y with zero

- To find the y-value of the vertex take the x-value of the vertex (AOS) and replace it with x.

- Vertex= (Axis of Symmetry, Optimal Value)

- GCF factoring is done by finding the GCF (cheak for GCF of an equation before factoring)

- Simple factoring is when a- value is equal to 1

-Complex factoring is when a- value isn't equal to 1

- Their special products called: difference of square and perfect squares trinomial

Learning Goals

1. Knows the importance of all variables in factored form (ex. a, r, and s values )

2. Can graph a parabola from a factor form equation

3. Knows and understands how to expand and simplify expressions

4. Can slove all 7 type of factored form equation

5. Knows the difference between simple and complex trinomial equations

6. Can identify and solve the special cases; difference of square and perfect square trinomial

7. Is able to understand and solve application problems

What is Factored Form?

Quadratic functions can be written in 3 different ways, vertex form, factored form or standard form. We are going to look at factored form. The factored form equation gives us the x-intercepts of the parabola, x-intercepts are also called zeroes or roots.


The factored form relationship equation is: y=a(x-r)(x-s)


The “r” and “s” are the values of the zeroes, which can be used to help find the axis of symmetry, the optimal value and the vertex of a parabola.

Overview : X-intercepts and Y-intercepts

Y-intercepts

To find the y-intercept replace x with zero


Example: y=3x^2+4x+1

y=3(0)^2+4(0)+1

y=0+0+1

y=1

*For y-intercept you can simply look at the c value of the equation


X-intercepts

To find the x-intercept replace y with zero

Example: y=3^2-6x

0=3x (x-2)

3x=0 or x-2=0

x=0 x=2


Axis of Symmetry

To find the axis of symmetry (AOS) take the two x-intercepts and add them, then divide by 2.

Example: (0+2)/2

=2/2

=1


Y- Value

-To find the y-value of the vertex take the x-value of the vertex (AOS) and replace it with x.

Example: 3x^2-6x

AOS= 1

=3(1)^2-6(1)

=3(1)-6(1)

=3-6

=-3

Vertex= (1,-3)

Zeros & X-intercepts

To find the zeroes of the parabola we have to first look at the equation, specifically the given zeroes.


Step 1: Make one side of the equation equal to zero. If an integer is replaced with the y- value subtract the integer from both sides, so one side of the equation is equal to zero.

*From here you technically ignore the a-value


Step 2: The “r” and “s” value in factored form needs to be taken out of the brackets. Then you make r and s value equal to x.

*In most cases one x- value will be positive and one will be negative.


Step 3: After you determine the x-intercepts, you write them in coordinates. For the example below, the parabola’s x-intercepts are at (6,0) and (-5,0).

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Extra Examples with solution
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Axis of Symmetry

The axis of symmetry is the x- value of the vertex. To find the axis of symmetry from factored form, we fist need o find the zeroes of the parabola and, then substitute the zeroes in the axis of symmetry equation.


Step 1: Find the zeroes of the factored from function


Step 2: Substitute the zeroes for the “r” and “s’ values and solve for the axis of symmetry.


Step 3: After you determine the axis of symmetry, write it in a coordinate.

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Optimal Value

The optimal value is the y-coordinate of the vertex, the high point on a parabola. When finding the optimal value of the parabola, you have to sub in the axis of symmetry of the parabola into the original factored form equation to solve for y or optimal value.

Step 1: Find the axis of symmetry


Step 2: Substitute axis of symmetry into the original equation to solve for y, which is the optimal value of the parabola.


Step 3: Write the optimal value in a coordinate.

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Vertex - is the highest point on a parabola. The x- coordinate of the vertex is of symmetry and coordinate of is the optimal value.

Vertex= (Axis of Symmetry, Optimal Value)

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What does this give you?

You have 3 points of the parabola: y=5(x-6)(x+5). You now have the vertex: (0.5, -151.25); 1st zero: (6,0) and 2nd zero: (-5,0).

You can use theses points to graph the parabola.

How do I make the Parabola ?

You just plot the 3 points and draw a u-shaped line through the 3 points, BOOM you have an outstanding parabola .

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The parabola: y=5(x-6)(x+5) will look like this.
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Expanding and Simplifying

A binomial is a polynomial with two terms. When multiplying two binomials together, you expand and simplify, using the FOIL method. The acronym FOIL stands for: First Outside Inside Last.


Step 1: multiply the first terms by inner term

Step 2: Multiply the first term by the last term

Step 3: Multiply the outer term by the inner term

Step 4: Multiply the outer term by the last term

Step 5: Simplify; add like terms

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Types of Factoring

1. Monomial common factoring (GCF)

Monomial common factoring is when an expression has a common factor in all of its terms. You must remove the common factor by dividing all the terms, this is necessary to get the correct factors.

Example : 8x- 8y

= 8(x-y)

answer: 8(x-y)

Step 1: Find GCF, which is 8

Step 2: Divide each term by 8

Step 3: Simplify and write the answer

2. Binomial Common Factoring (GCF)

In Binomial Common Factoring we have to find the GCF to factor the expression.


Example: 3x (x+5) -2 (x+5)

= (x+5) (3x-2)

ans: (x+5)(3x-2)

Common Binomial Factors

3. Factoring by grouping (4 terms)

Factoring by grouping has 4 terms. You group the first two and last two terms, then factor out a GCF for each group.

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Factoring by Grouping - MathHelp.com- Algebra Help

4. Simple trinomial factoring

When factoring a simple trinomial, the a- value is always 1. In order to factor the equation, you need to find 2 numbers that is the product of c, the last term, and the sum of b, the middle term.

Step 1: Find two numbers that have a product of c, and a sum of b

Step 2: Insert 2 parentheses each with x and one of the 2 numbers that was used to produce c and add up to b

Step 3: After you factor it, check your answer.

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Simple Trinomial Factoring

5. Complex trinomial factoring

Complex trinomials have an a-value that is not equal to 1. In order to factor the equation you have to find two numbers that is a product of ac, and a those numbers need to have a sum of b.

Step 1: Decompose the middle term. Find two numbers that multiply to “a” and add up to “b”

Step 2: Common factor the 1st two terms

Step 3: Common factor the last 2 terms

Step 4: Common factor the step 1 and step 2 to get the final factors

Step 5: Check your answer

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6. Special Product- Difference of Squares

These are binomial equations where the sign in the middle must be negative. You simply square root the first term and the second term and you write down the factors with opposite signs. Differences of squares produce the sum and difference of the square root.

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7. Special Product- Perfect Square Trinomial

These are trinomials where the first and last terms are perfect squares and the middle term is twice the product of square roots of the first and last terms.


Step 1: Check if the first and last term are perfect squares (you do this by squaring the terms)

Step 2: Write the product of the square of the 1st and last term in parentheses and then square it.

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Example 3: Factoring quadratics as a perfect square of a sum: (a+b)^2 | Algebra I | Khan Academy

Word Problems in Factored Form

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Quadratics in Standard Form

Summary

  • Standard form is written as y=ax^2+ bx+ c
  • The “c value” is the y-intercept of the parabola
  • Complete the square to find the vertex. This method turns the function standard form to vertex form.
  • If the “a- value” is more than 0 the parabola opens upwards
  • If the “a- value” is less than 0 the parabola opens downwards
  • Use the quadratic formula to find the zeros. Substitute the “a- value”, “b-value”, and “c-value” into the quadratic formula equation to find the zeros of the function
  • Substitute the “a- value”, “b-value”, and “c-value” into the discriminant formula equation to find if the function has 0, 1, 2 solutions. If the solution is positive there are 2 zeros, if solution is negative there is 0 zero, if solution is 0 there is 1 zero
  • Use the zeros, the vertex, y-intercept to graph the quadratic function
  • If y-coordinate of the vertex is positive its Maximum value
  • If y-coordinate of the vertex is negative its Minimum value

Learning Goals

  • You can find the y-intercept of the parabola
  • Can find the zeros of the function
  • Can solve to find the vertex of the function
  • Knows how to complete the squares
  • Knows how to find zeros with using the quadratic formula
  • Can solve for discriminant to figure out if the function has 0, 1, or 2 zeros
  • Can confidently claim if a function has a maximum or minimum value
  • Can use the zeros, the vertex, y-intercept to graph the quadratic function

Completing The Square

Steps :

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Example 1:

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Example 2:

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Converting Standard to Vertex Form

Quadratic Formula

Steps:
  1. Identify the a, b, and c value. Arrange terms if necessary
  2. Make the equation equal to zero
  3. Clearly identify the “a- value”, “b-value”, and “c-value”
  4. Substitute the “a- value”, “b-value”, and “c-value” into the quadratic formula
  5. Solve and simplify
  6. Clearly write the zeros in the simplest form. Final zeros need to be in coordinates
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Solve Quadratic Equations using Quadratic Formula

Axis of Symmetry

Formula x=-b/2a

Step 1: Substitute the variables

Step 2: Solve and simplify


Formula: x= r+s/2

Step 1: Substitute the zeros for the “r” and “s” values

Step 2: Divide

Step 3: solve and simplify

Vertex

Step 1: Substitute the axis of symmetry into the equation for the “x- value”

Step 2: Solve and simplify

Step 3: Clearly identify the vertex in coordinates

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Maximum or Minimum Of A Quadratic Function

The y- coordinate of the parabola's vertex can either be the maximum or minimum value.

NOTE

  • If the “a- value” is positive the parabola opens upwards
  • If the “a” is negative the parabola opens downwards
  • If y-coordinate of the vertex is positive its Maximum value
  • If y-coordinate of the vertex is negative its Minimum value
Finding the Minimum or Maximum of Quadratic Functions

Discriminant

Note:

  • When discriminant is positive there will be 2 solutions to the parabola
  • When discriminant is negative there will be 0 solutions to the parabola
  • When discriminant is 0 there will be only 1 solution to the parabola
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Discriminant of quadratic equations | Polynomial and rational functions | Algebra II | Khan Academy
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Standard Form Word Problem

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Relationship

Vertex Form and Graphing, + Other Relations

  • Graphing is easier in vertex form because the vertex is already given so you just have to find the 2 other points

  • Can easily tell if the graph is compressed or stretched by the a- value of the equation

  • Automatically know the axis of symmetry and optimal value of the function which is the x and y coordinate of the vertex

  • Can change into standard form when expanded, and then can also be turned into factored form by factoring or doing the quadratic formula which will also include finding the vertex and the a-value to write the function in factored form

Standard Form and Graphing + Other Relations

  • Really easy to tell when the parabola crosses the y- axis also known as the c value in the equation, this can be very helpful for word problems

  • Completing the square gives the vertex, axis of symmetry, and optimal value because it turns the function into vertex form

  • With vertex, axis of symmetry,optimal value and y-intercept we can easily graph the parabola.

  • This form can turn into factored form by factoring if its “possible”. If not you can also use the quadratic formula to find the zeros, and then the a-value to turn standard form into factored form.

Factored Form and Graphing + Other Relations

  • Factored form allows you to clearly see the zeros of the parabola

  • Finding the zeros is very helpful in finding the axis of symmetry and optimal value, and vertex

Can successfully graph the quadratic functions with all these points.

  • To find the optimal value of a function we replace the x/ the value of the axis of symmetry into the equation and solve for the optimal value, also known as the y-coordinate of the vertex. This method was taught and used in vertex form.

  • Factored form can change into standard form by expanding, then from standard form it can turn into vertex form by completing the squares

Reflection

Quadratic was a long, 3 parted unit. Quadratics was one of my favorite units. It is not because it was easy, it was because it was very enjoyable. Anyone can perform well in this unit if they try their best and practice, practice and practice. That’s how I did well in the unit. The quadratics unit was split into 3 units: vertex form, factored form, and standard form. All of these topics require a good understanding of linear relation from grade 9. Quadratics in standard form was the easiest because there were only 2 main concepts we had to learn, quadratic formula and completing the squares. In the quadratic formula, we just have to substitute the a,b, and c values into the formula and solve to find the zeros, which was very easy. When “completing the square” you need to know the concepts of how the factored form since it requires some factoring and knowledge of perfect square trinomials. Factored form was also very simple, but the most important skill you require in this unit is strong arithmetic skills. Vertex form is also very simple but I really needed to pay special attention to the signs of the function. For the vertex form test, I alway forget to put the ^2 in the formula

y=a(x-h)^2+k, that's where I lost some of marks.

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My best test for quadratics was for standard form, I achieved a mark of 38/41. I was very strong for the application and knowledge portion but I lost 3 marks for communication because I didn't clearly explain my answer. Note: never forget to write "let statements" for word problems. My weakest category is thinking. The TIPs assignments just drain my thinking mark. I find the thinking question a little bit confusing, so I often over think them, which makes me stressed and results in poor marks on the TIPs assignments. Other than the some TIPs assignments, the unit of Quadratics is very easy to do well

if you put in the efforts to do well in it.