By: Sharan M

## Second Differences

Going back to grade 9, students have been taught about linear relationships and first differences. Grade 10 introduces students to quadratic relationships which can be found by checking the second differences.

## Definitions

Here are some definitions one should know before starting Quadratics,

1. Parabolas: We have seen lines on a graph, when a curve is formed on a graph it's called a parabola.

2. Vertex: Vertex is the highest point on a graph.

3. Zeros: Zeros are basically the x intercepts. The x intercept is the point where the line touches the x axis on a point. The x intercepts/ zeros are also the x value of the vertex.

4. Optimal Value: Optimal value is the y value of the vertex.

5. Maximum Value: When any equation/ word problems tells us to find the maximum value of something, they are referring to the vertex.

## Three types of equations

There are three types of equations used in the quadratics unit.

- The first type of equation is written in vertex form. y=a(x-h)²+k is the basic equation written in vertex form.

- Next equation is written in factored form. y=a(x-r)(x-s) is the basic equation.

- The last type of equation is called standard form. y=ax²+bx+c is the equation we can use to go from vertex to factored form.

## Parabolas/ Vertex form

The equation y=a(x-h)²+k is used to determine how the parabola moves.The equation is in vertex form.

The a value in front of the equation tells us whether the parabola opens up or down. If the a value is a positive whole number of a fraction, the parabola will open up. When the a value is a whole number, its called a vertical stretch. If the a value is a fraction, its parabola's transition is described as a vertical compression.

The picture below shows the shape of the parabola when it opens up. An easy way to remember this is the shape of a U.

If the a value is a negative whole number or a fraction, then the parabola opens downward.

An easy way to remember when a parabola opens down is when the parabola forms a downward U shape.

The h value in the equation represents the x value of the vertex. Knowing the x value of the vertex is important specially when we have to graph the parabola. The h value moves the parabola left or right.The h value is a horizontal shift by a factor of the number/fraction. When an equation is given and we have to find the h value, the number is always opposite. For example,y= (x - 3)² + 4. In this equation there's a negative sign in front of the 3 but since its the opposite operation we use, the value of h will be a positive three. The k value of the graph is the y value of the vertex, and is also called the optimal value. This value moves the graph up or down. The k value represents the vertical shift of the parabola.

## Graphing Parabola

To graph a parabola, we have to graph the step pattern first. The step pattern is 1,4,9 which comes from the square roots of 1,2 and 3.
How to Graph Parabolas

## Writing Equations

When given the vertex and an x intercept, we can come up with an equation for it in vertex form. If the vertex is (1,3) and the x intercept is ( 2,0), we can solve like this. Remember to use the vertex form equation and plug numbers from there.

## Graphing from Factored Form

There is another way of graphing. We can graph the parabola from an equation thats written in factored form.

y=0.5(x+3)(x+5) is an example of an equation written in factored form.

To graph a parabola, we need to find three peices of information.

1) State the zeros. ( This can be found by setting each factor equal to zero)

2) Find the axis of symmetry. ( This is where we add the two values of zero and divide the sum by 2 since we are finding the midpoint)

3) State the optimal value.( This is where we sub in the value of the axis of symmetry into the equation).

1. y=0.5(x+3)(x+5)

x+3=0 x+5=0

x=0-3 x=0-5

x= -3 x=-5

Therefore, the two x intercepts are (-3,0) and (-5.0)

2. Axis of symmetry= (-3)+(-5)/2

= -8/2

= -4

Now we found the x value of out the x value of the vertex which is (-4,0)

3. y=0.5(x+3)(x+5)

y=0.5( -4 +3)( -4 +5)

y=0.5(-1)(1)

y= -0.5

Now that we found the y value of the vertex (-0.5), we can graph it.

## Parabola Intercepts.

A parabola can have two x intercepts or one x intercepts.

When a parabola has 2 x intercepts, it touches two points on the x axis. The points can negative or positive. When a parabola only has one x intercept, its the vertex that's the x intercept. A parabola can also have no x intercepts, this is when the vertex is above or below the x axis.

## Expanding

Expanding is one of the things one needs to know in order to learn factoring. We are given expressions in factored from and we have to expand it. We basically multiply the terms inside the brackets with terms outside the bracket.

(x+2) ( x+5) is an example.

1. We can draw arrows to help us. Draw an arrow from the first x to the second x in the bracket. Next draw an arrow from the first x to 5 in the second bracket. When the multiply the two x's together, the answer is x² since there are two x's. When x and 5 are being multiplied we get 5x as the answer.

2. We also have to look at the second number in the first bracket which is 2. Multiply 2 by x and by 5. After multiplying these numbers, we get 2x and 10.

3. Join the answers together.

(x+2) ( x+5)

= x²+ 5x+2x+10

4. From this answer, check to see like terms and solve.

(x+2) ( x+5)

= x²+ 5x+2x+10

= x²+7x+10

5. x²+7x+10 is the final answer as there are no like terms.

In the example, arrows are shown so that you can see which numbers are multiplied together.

Remember we can have any letter in front of the number, and in this example, two n's are multiplied together so it creates n².

The video explains expanding with more practise questions
Multiplying Out Two Brackets

## Factoring Video

This is a factoring video we made that explains all types of factoring problems and also gives examples of each one.

## Complete the Square

In equations when one has to complete the square, its important that one knows the square roots. We use complete the square when we have to find the maximum and minimum value (vertex). These types of equations are given in standard form and we can find the vertex form by using complete the square. This method is done differently.
In the above example, the 4 is written with a positive and a negative sign because it can be positive or negative. Its a 4 because we take the half of the 4 in front of the x which is a 2, and then we square root it, which is a 4. In the third step, the negative four is moved out because we only want to keep the perfect square section in the bracket. Then we solve (-4)- 13 and square root inside the bracket. Then we can branch off to the two possible solutions, which can be negative and positive.

We can solve an equation given in standard form by using the Quadratic Formula. We dont have to convert the equation in vertex form if we know the Quadratic Formula. The Quadratic formula can be used when we cannot factor.

The reason why there are two signs ( negative and positive) after the b value is because we can have two x intercepts, meaning two solutions.

To use the Quadratic formula, one can simply sub in numbers where they need to go. Then one can solve the equation.

Example: 9x²-24x+16=0

## Discriminants

Discriminants come from the Quadratic Formula. To find a discriminant we use part of the Quadratic formula which is b²-4ac. Discriminant is a quick way to check how many x intercepts there will be. The discriminant is the number that needs to be square rooted.Rather than doing the whole quadratic formula, the discriminant formula is short and quick.

Example#1 : 0= 5x²+3x+2

Formula: b²-4ac

=(3)²-4(5)(2)

=9-40

= -31 Therefore the -31 is the number inside the square root, and since we cannot square root a negative number, there is no solution to this equation meaning no x intercept.

This formula comes in handy when you have to graph a parabola. By checking the discriminant, one can already know how many x intercepts there will be.

Example #2 : y= 2x²+5x+3

Formula: b²-4ac

(5)²- 4 (2)(3)

25-16

9

Therefore 9 is the number inside the square root and we know that this equation will have two solutions because the 9 can be positive or negative.

The problem above asks us to find the area of the shaded figure. We know that the little square on the inside and outside have the same dimensions. This means that to get the area we basically multiple the one dimension given. Then we can subtract the two areas to get the area of the shaded region.
To solve the above problem, first find the area of the big square and the small square. Big square will be called figure 1 and small square will be called figure 2.

Area of figure 1: ( 3x+4) ( 3x+4) Step 1 : Expand to multiply the expressions

9x²+12x+12x+16 Step2: After solving, collect like terms

9x²+24x+16 This is the area of figure 1.

Area of figure 2: (x-5) (x-5) Step 1 : Expand to multiply the expressions

x²-5x-5x+25 Step 2 : Collect like terms from this point.

x²-10x+25 This is the area of the figure 2

To find the area of the shaded region, all we do is subtract figure one's area with figure 2.

(9x²+24x+16) - ( x²-10x+25 )

= 8x²+34x-9

Remember to only subtract the like terms together

Write a conclusion statement

Therefore, the area of the shaded region is 8x²+34x-9

This is another type of motion word problem.
Solving the "Ball thrown in the air" problem

## Revenue Word Problems

These are one different types of word problem. They give a regular price which is included in the expressions and the word problems also increase or decrease the amount of something thats being talked about in the work problem. Similarly, a revenue word problem is ,

## Connections

Quadratics is a long unit, but everything is linked to something else. First we learn expanding because we should be able to expand while doing factoring problems. If one doesn't know expanding, they wont know factoring either.

There are also multiply ways to graph a parabola, we can use vertex form, standard, or factored form equation. One can go from standard to vertex and to the factored form easily. The easiest way to graph can be using the factored form equations because we can find the x intercepts but setting each expressions equal to 0, then find axis of symmetry and the optimal value. One can also go from the vertex to factored form and then graph.

There are also multiple ways to solving an equation. Any types of factoring, expanding or the quadratic formula can be used.

We can also find the vertex by solving the equation by completing a square.

Even in word problems, one might have to factor and then use the quadratic formula.

Everything is linked and fits into pieces! Always look for the multiple ways to solve any problem

The vertex of the above equation is (-2,1) because the equation is in the vertex form and we know the h value is the x value of the vertex. The h value is positive two in the equation and sine we always write the h value opposite, the vertex is -2. The k value of the equation is 1 which means its also the y value of the vertex.

## Personal Reflection

This is a picture from one of the quizzes and the question had an equation given in the standard form which had to be graphed. To graph, we need x intercepts and I was able to factor the equation and find the x intercepts. I also had to find the vertex to graph. However, in the quiz I only found out the x value of the vertex and I forgot what to do with it. Clearly, the point had to be graphed but I also needed to find the y value of the vertex which I didn't find on the quiz. I basically forgot about plotting the x and y value of the vertex so I couldnt graph the parabola. Looking back now, I remember that to find the y value of the vertex, all i had to do was sub in my x value to the original equation. The y value would have came to be -8. After this, all I had to was plot (3,-8) and that was the vertex. I forgot this part of the quiz because I didn't review it before, and I learnt the lesson!