By Raghav Sharma
UNIT 1: Intro into Quadratics
- Parts of a Parabola
- First and Second Differences
- Vertex Form
- Word Problems Using Vertex Form
- Graphing Transformations
- Mapping Notation
How calculate first and second differences
Know the vertex form
Learn how to solve word problems using the vertex form
Know all the transformations
Learn mapping notations
Features of a Parabola
Vertex: The highest or lowest point on a parabola (written as a coordinate pair)
Axis of symmetry: The vertical half point that separates the parabola into 2 equal parts (written as only a number on the x-axis).
Optimal Value: Can either be Minimum or Maximum value. If parabola opens up, there will be a minimum value, and if the parabola opens down, there will be a maximum value (written as only a number on the y-axis)
Zeroes: Also known as roots, are points on the graph where the parabola touches or crosses the x-axis.
FIRST AND SECOND DIFFERENCES
1. The a value is the parabola's vertical stretch
- If a is a negative number, the parabola is flipped (opens downwards) or in other words, there is a reflection in the x-axis
2. The h value is the parabola's horizontal translation
- If h = 3, the parabola's axis of symmetry (the middle point) would be at -3 on the x-axis
- If h = -7, the parabola's axis of symmetry would be at 7 on the x-axis
3. The k value is the parabola's vertical translation
- If k = 3, the parabola's optimal value would be at 3 on the y-axis
- If k = -10, the parabola's optimal value would be at -10 on the y-axis
4. By looking at both the h and k value, you can combine both of them to determine the vertex of the parabola
-If the h value is -5 and the k value is 9, the vertex is (5, 9)
GRAPHING USING VERTEX FORM
1. As mentioned above, the h value, which is always opposite (negative value is positive on the graph and vice versa) tell us the x value of the vertex
2. The k value, which is NOT opposite, tell us the y value of the vertex
3. The a value will tell you the parabola's vertical stretch. If it is negative, the parabola opens downwards. To use the vertical stretch, you must use the step pattern (one left/right and one up/down. two left/right and four up/down) and multiply the number on the up/down. (e.g. A vertical stretch of 2 is one left/right and two up. two left/right and eight up)
- To go from vertex form to the equation, first find the vertex's x and y value.
- The x value is the h. Make sure to make it is negative if it's positive and vice versa
- The y value is the k. This one is not opposite
- To find the a value, count once right or left of the vertex and count how many times you have to go up or down until you reach a point where the parabola intersects with both the yand x axis lines. This number is your a value. Your a value will be negative if the parabola opens downwards
- Finally, input the values in the basic equation and you have the answer
When graphing the vertex form there are 4 important things to look for in the given equation:
- Vertical Stretch/Compression
- Reflection in the x-axis
- Horizontal Translation
- Vertical Translation
1. Vertical Stretch/Compression
The vertical stretch/compression in a quadratic equation determines whether the parabola contains wide curve or a narrow curve. When a > 1 the parabola has a vertical stretch. When a negative sign is placed in front of it, the rule still applies however it is a < -1. When
0 < a > 1 the parabola has a vertical compression. Just like a stretch, when a negative sign is placed in front of it, the rule still applies but between 0 and -1.
2. Reflection in the x- axis
If the a value given in the quadratic equation has a negative sign in front of it the parabola is reflected into the x-axis meaning it will have a downwards opening. On the other hand, if the a value in the given equation has a positive sign in front of it the parabola will have an upwards opening.
3. Horizontal Translation
The horizontal translation is the variable h in the vertex form. The h value determines where the x in the vertex lies. If the h value is written as a negative in the equation it will be placed on the graph as a positive, and if the h value is written as a positive in the equation it will be placed on the graph as a positive. When h is a negative value the translation on the graph is to the left, and when h is a positive value the translation on the graph is to the right.
Example: y = -1 (x-1)^2 + 1
- The h value in the equation is -1, meaning on the graph it will be placed as +1
- This is a horizontal translation 1 unit right
4. Vertical Translation
The vertical translation is the k value in the vertex form. The k value determines where the y in the vertex lies. When the k is a positive the y value moves up and when it is a negative the y value moves down. For example, if the k value is positive 7, the y value in the vertex would be positive 7 and this would be a vertical translation 7 units up.
Example: y = -1 (x-1)^2+1This is a vertical translation 1 unit up
The Example below shows how the vertical translation effects the placing of a parabola on a graph.
Now, for example, if we were to describe the transformations for the parabola
y = 4 (x - 6)² - 8, they would be:
- parabola opens upwards
- there is a vertical stretch by a factor of 4
- there is a horizontal shift to the right by 6 units
- there is a vertical shift down by 8 units
- vertex = (6, -8)
What really happens to the key coordinates of y = x² to become y = 4 (x - 6)² - 8 :
(x, y) ⟶ (x + 6 , 4y - 8)
(x, y) ⟶ (x - h, ay + k)
- y = (x - 4)² + 2 (x, y) ⟶ (x + 4, y + 2)
- y = (x +7)² (x, y) ⟶ (x - 7, y)
- y = 5 (x + 2)² - 3 (x, y) ⟶ (x - 2, 5y - 3)
- y = ½ (x - 3)² + 1 (x, y) ⟶ (x + 3, - ½y + 1)
- y = - 2x² + 6 (x, y) ⟶ (x, - 2y + 6)
UNIT 1: FACTORED FORM
- Expanding & Simplifying
- Factored Form
- Factoring common factors
- Factoring Common Binomials
- Factoring by Grouping
- Special Cases
- Perfect Squares
- Difference of Squares
- Factoring Simple Trinomials
- Factoring Complex Trinomials
- Solving word problems
- Learn how to expand and simplifly
- know the factored form
- learn hot to get factored form
- learn how to factor binomials
- know how to factor by grouping
- learn special cases
- know how to factor simple trinomials
- know how to factor complex trinomial
- learn how to solve word problems
EXPANDING AND SIMPLIFYING
The values of r and s are the x-intercepts/ roots/ zeros and are written like: (r, 0) (s, 0)
The value of a will tell you the shape and direction of the parabola. Just like in vertex form, the a value determines the stretch or compression of the parabola (how wide or narrow it is). If the value of a is positive, the parabola will open upwards. If the value of a is negative, the parabola will open downwards (reflect onto the x-axis).
Common factoring is the opposite of expanding. Expanding involves multiplying while factoring involves dividing.
Different ways to Factor :
- Finding the GCF
- Factor by Grouping
Finding the GCF
When using this method, find the greatest common factor of the polynomial's terms. This will include the GCF of its coefficients and the GCF of its variables.
In this example, the GCF of the coefficient was 2 and the GCF of its variables was y^2
FACTOR COMMON BINOMIALS
Example: 3x (z - 2) + 2y (z - 2)
In order to factor an equation like 3x (z - 2) + 2y (z - 2), you must think of (z - 2) as one factor. Therefore, you would take out (z - 2) and write it outside the bracket, and write what's left on the inside.
Solution: z - 2 (3x + 2y)
Therefore, a binomial can also be considered a common factor.
Example: -8x (z + 4) - 5y (z + 4)
Solution: (z + 4) (-8x - 5y)
FACTOR BY GROUPING
For example: df + ef + dg + eg
Split the equation into two halves (be sure to keep the middle sign with the middle term).
df + ef / + dg + eg
- Group the first two terms together and the second two terms together by placing brackets around them. NOTE: always put a plus (+) sign in between the two brackets and if there is a minus (-) sign, put it on the inside of the bracket.
(df + ef) + (dg + eg)
- f is the common factor in the first set of brackets, and g is the common factor in the second set of brackets.
f (d + e) + g (d + e)
- Notice that what is left in the brackets is the same. Therefore, (d + e) is a common factor.
(d + e) (f + g)
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
To check if the equation is a perfect square, you simply confirm that you can square the first and last term nicely (resulting in a whole number, not a decimal). Next, you must multiply the value of the first term by the value of the last term and multiply by two. If this gives you the middle term, the equation is a perfect square. If this does not give you the middle term, the equation is not a perfect square and you must try a different factoring method. Your answer is written as either: (a + b)² or (a - b)², depending on if the middle term is negative or positive.
EXPAMLE 1: x² + 12x + 36= (x + 6) (x + 6)
= (x + 6)²
This solution fits the perfect square pattern since both x² and 36 are perfect squares, and 12x is twice the product of x and 6. Since all signs are positive, the pattern is a² + 2ab + b² = (a + b)².
Difference of squares will usually follow the pattern: a² - b² = (a + b) (a - b)
And will NEVER follow the pattern: a² + b²
This equation shows a formula for factoring a² - b², the difference of two perfect squares. Notice that the factors are identical except that one is addition and the other is subtraction. An easy way to identify if an equation is a difference of squares is to count the number of terms it has. If the equation only has 2 terms, it is difference of squares.
- x² - 64
= (x + 8) (x - 8)
This solution fits the difference of squares pattern since both x² and 64 are perfect squares and the problem is subtraction. Since this is the case, the formula for difference of squares can be used: a² - b² = (a + b) (a - b).
FACTOR SIMPLE TRINOMIAL
We use the method of factoring simple trinomials to change an equation in standard form [ax² + bx + c] to factored form [a (x - r) (x - s)] .The value of c is a product of two numbers that also add up to give you the value of b. For example, in the equation x² + 7x + 10, the numbers 5 and 2 multiply to 10 (c) and add to 7 (b).
Another way to write this equation is:
x² + (m + n) x + mn
= (x + m) (x + n)
Using the example above:
x² + 7x + 10
= (x + 5) (x + 2)
Solving Using The Guess & Check Method:
- Look for and remove the common factor (if any)
- Choose factors for the constant term (c) that sum to the x term (b)
- Check by expanding (recommended)
- n² - 10n - 16
= (n - 8) (n - 2)
- z² + 8z + 7
= (z + 7) (z + 1)
- t² + 11t - 26
= (t + 13) (t - 2
FACTOR COMPLEX TRINOMIAL
SOLVING WORD PROBLEMS
- Standard Form Equation: y = ax² + bx + c
- a value gives you the shape and direction of openingof the quadratic
- c value gives you the y-intercept of the quadratic
- To get x-intercepts, SOLVE using the "Quadratic Formula"
- MAX or MIN? Complete the square to get vertex form.
- Discriminant Formula: x=(b^2 - 4ac).
- In order to find the x-intercepts use the Quadratic Formula.
1.Quadratic Formula x-intercepts.
2. I will know how to use the Discriminant to find how many x-intercepts a quadratic function has.
3. I learned how to graph a Quadratic Function using the Quadratic Formula.
4. I learned how to Complete The Square to find the Maximum/Minimum value of the quadratic and its' vertex.
5. I will know how to solve a word problem
GRAPHING STANDARD FORM
example of y=3x²+6x-2 graph
COMPLETING THE SQUARE
- y = ax^2+bx+c
And the vertex form equation looks like:
- y = a(x-h)^2+k
This part of the website will teach you how to go from standard for to vertex form.
- The first step is to place a bracket in which the a becomes isolated
- Move the c value outside of the brackets
- Divide the b value and square it. This becomes your new c value inside the bracket
- Add the opposite of the c value outside of the brackets
- Factorize the bracket
- Evaluate the outside term (Remember to apply the a value to the inverted c value)
How to Solve A Completing the Square Word Problem
Starting slow in the beginning i have made big progress and I know understand the three main components of these units.
- Goes into Factored Form by finding the axis of symmetry and the vertex. Then sub in the vertex into the vertex form equation.
- By completing the square, you will be able to convert Standard Form into Vertex Form
- By solving for the zeroes in Vertex Form, you could convert it into Factored Form aswell
- Standard Form can be changed into Factored Form by using the 6 different methods of factoring ( Perfect Square and or Trinomial)
- Both Factored and Vertex Form can be changed into Standard Form by expanding and simplifying the equation