Quadratic Relationship
By, Drasti Patel
Why are Quadratic Relations important?
The Path of this basketball is a parabola
Rollacosters are made up of Parabolas
This bridge is made with the help of quadratics relations
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.
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?
Parent Function
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.
Vertex
Optimal Value
Maximum and Minimum Value
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
Unit 1: Graphing in vertex form
Summary Of Graphing in Vertex Form
- 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?
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
Vertical Shift
Verticleal stretch & compression
Horizontal Shift
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.
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.
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.
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.
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.
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.
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.
(x,y) ------> (x+3), (2y-4) because the mapping formula is x+h, ay+k .
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.
Word Problems in vertex form
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).
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).
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)
- What is the initial height of the baseball?
- What is the maximum height?
- What is the height of the ball after 2 seconds?
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.
Quadratics in Factored Form
Summary of Quadratics in Factored Form
- 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
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).
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.
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.
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)
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 .
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
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)
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.
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.
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
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.
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.
Word Problems in Factored Form
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 :
Example 1:
Example 2:
Quadratic Formula
- Identify the a, b, and c value. Arrange terms if necessary
- Make the equation equal to zero
- Clearly identify the “a- value”, “b-value”, and “c-value”
- Substitute the “a- value”, “b-value”, and “c-value” into the quadratic formula
- Solve and simplify
- Clearly write the zeros in the simplest form. Final zeros need to be in coordinates
Axis of Symmetry
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 2: Solve and simplify
Step 3: Clearly identify the vertex in coordinates
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
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
Standard Form Word Problem
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.
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.