By: Nargiz M
Table of contents
- Introducing the Parabola.
- Analyzing quadratics
- Graphing from vertex form
- Finding the equation given vertex
- Graphing from factored forming
- Common factoring
- Factoring by grouping
- Factoring simple trinomials
- Factoring complex trinomials
- Factoring special trinomials
- Completing the Square
- Solving by isolating x
- Quadratic Formula
- Shape problems
- Revenue problems
- Motion problems
- Consecutive intergers
What is Quadratics?
Lesson #1: What is a Quadratic Function?
The graph of a Quadratic Function is a curve better known as a Parabola. Parabolas may open upward or downward and vary in width and steepness. However, they all have the same basic curved “U” shape. The picture attached below show four detailed graphs, all of which are Parabolas.
All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point known as a vertex of the parabola. If the parabola opens up, then the vertex is the lowest point, and is known as the minimum point. In both, the vertex is the optimum value. – the minimum and maximum points on a graph- If the parabola opens down, the vertex is the highest point. This point is called the maximum point. A parabola also contains two points called the zeros better known as the x-intercepts
Table of Values to analyze Quadratics
Lesson #2: Using a Table of values to analyze Quadratics
As previously learned, Linear systems are straight lines, Linear relations the first differences are always the same, however in Quadratic Functions the second differences are the same.
A quadratic function can be graphed using a table of values. The graph creates a parabola. The parabola contains specific points, the vertex, and up to two zeros or x-intercepts. The zeros are the points where the parabola crosses the x-axis.
If the coefficient of the squared term is positive, the parabola opens up. The vertex of this parabola is called the minimum point. Example : y = a(x-h) + k
If the coefficient of the squared term is negative, the parabola opens down. The vertex of this parabola is called the maximum point. Example : y= -a (x + h) +k
To help you graph, Technology can be very helpful, try using desmos , to help you graph parabolas.
The equation of a parabola can be expressed in either; Standard form and Vertex form.
The vertex form of a parabola is expressed as; y = a(x-h)2+k
A, H, and K are all essential factors of the Parabola.
- If A is positive then the parabola opens upward
- If A is negative the graph opens downward
- If the value of A is greater than 1 then the parabola widens
- If the value of A is less than 1 then the parabola narrows down.
- When translating the graph, H determines whether the Parabola will move sideways
- The coefficient of H is always opposite of what it is. + = neg , neg = +
- When translating the parabolic graph, K determines whether the parabola will move up or down.
Here are the steps to graph a parabola only given the vertex form:
Find the vertex. Since the equation is in vertex form, the vertex will be at the point (h, k).
Find the y-intercept. To find the y-intercept let x = 0 and solve for y.
Find the zeros. To find the x-intercept let y = 0 and solve for x.
Graph the parabola using the points found in steps 1 – 3.
Y= a(x-h)^2 +K ---- (The vertex is (-2,8) and passes through the point (2,0)
Step One: insert all numbers into designated places. You will be solving for A.
y=a(x+2)^2 + k
y=a (x+2)^2 + 8
0 = a (2 +2)^2 + 8
0= a (4)^2 +8
0= a16 + 8
subtract 8 from both sides
-8 = a16
-8/16a = -1/2
= y= -1/2 (x+2)^2 + 8
Now choose points, using the table of values, and graph the Parabola
Lesson: #6 Factored Form
The factored form of a Quadratic question is : y= 0.5(x+3) (x-9)
Step 1 State the zeroes : The Zeros are found by setting each factor equal to zero.
: Set y=0
x+3 = 0 x-9=0
Step 2 Axis Of Symmetry: is the midpoint of the zeroes.
X= -3+9 = 6/2 = 3
Step 3 Optimal Value : is found by subbing the axis of symmetry value into equation
Y= 0.5 (x+3)(x-9)
0= 0.5 (3+3)(3-9)
=0.5(6)(-6) = -18 (Vertex)
The video attached will guide you into solving what is needed for this form.
Factoring By Grouping
Factoring Simple Trinomials
Factoring Complex Trinomials
But now… Complex trinomials starts with a number that is not equal to 1.
Remember to always find multiples of ax2 and C to get the middle term, also to common factor if necessary.
Factoring Special Trinomials
Perfect squares -
Step 1: Square root and find the multiples of a and c
Step 2: Then multiple the numbers by 2 to equal middle term
Step 3: equals to middle term = perfect square
Different of squares - ( x + z ) ( x - z )
Step 1: Square root and find the multiples
Step 2: Insert both the numbers you got in the brackets one + and the other -
Completing the Square
Solving by isolating x
This formula helps you find the X-InterceptsStep 1. Find the Quadratic Formula
Step 2. Substitute in the variables, So, plug in the values for a, b, and c
Quadratic word problems - Shapes
Answer below :
Quadratic word problems- Revenue
Answer below :
Example of a Consecutive integer problem :
Here is a video on Factoring to develop your understanding :
- Common Factoring
- Factoring By grouping
- Simple trinomials
- Complex trinomials
- Perfect/Different squares