Grade 10 Quadratics
Graphing, Factoring, Solving
By: Simran Gakhal
Table of Contents
*Quick Look Back to Quadratic Relations
1. Expanding and Simplifying
- Distributive Property with Factored Form
- Factoring Standard Form
-Simple Trinomials
-Complex Trinomials
-Difference of Squares
-Perfect Squares
-Common Factoring
- Completing the Square
2. Solving
- Solving Factored Form
- Solving Standard Form
- Solving Vertex Form
3. Graphing
- Vertex Form
- Vertex and Step Pattern
- Transformations
- Factored Form
- x Intercepts
- Axis of Symmetry
-Optimal Value
- Standard Form
- x Intercepts
- Axis of Symmetry
4. Word Problems
5. Reflection
6. Connections
Linear & Quadratic Relations
EXPANDING AND SIMPLIFYING
Distributive Property with Factored Form
Factored Form
Distributing
Distributing
Multiplying
Adding Like Terms
Final Product
Factoring Standard Form
Solving Complex Trinomials
Solving Perfect Squares
Solving Difference of Squares
Completing the Square
SOLVING
Solving Factored Form
Factored Form
Zeros & Axis of Symmetry
Optimal Value
Solving Standard Form
Quadratic Formula
Discriminant
X-Intercepts
Solving Vertex Form
Vertex Form
Y-Intercept
X-Intercept
X-Intercept
GRAPHING
Graphing Vertex Form
It is very easy to graph a vertex form equation because most all of our information is seen in the equation. A vertex form equation looks like y = a(x-h)²+k
- The 'a' tells us the stretch/ compression of the graph
- The 'h' tells us how many units left/right the vertex is going to move from 0
- The 'k' tells us how many units up/down the vertex is going to move from 0
- The 'x' and the 'y' are coordinates on the graph
One key point you need to know is that (h, k) are the vertex. This is why it is easier to graph a vertex form equation because the vertex is given in the equation.
You also need to know that when your parabola is going to open upwards or downwards. This information is also found in the equation. If the 'a' is negative then your graph will open downwards and if the 'a' is positive, it will open upwards.
STEP PATTERN
-2 ........... -4
-1 ........... -1
0 .............. 0
1 .............. 1
2 .............. 4
Basic Pattern:
Over 1 -- Up 1
Over 2 -- Up 4
Graphing Vertex Form
Positive
Negative
Transformations
Math has its own language and there is specific terminology that you must know. When explaining the transformations of certain coordinates of a vertex form graph, there are certain words we use. These words are:
- Vertical Stretch: If the 'a' is a number greater than 1 then the graph would have a vertical stretch which means that is steeper. The way you would write it: This graph has been vertically stretched by the factor of 2 (or any number greater than 1).
- Vertical Compression: If the 'a' is a number less than 1 then the graph would have a vertical compression meaning it would be wider. The way you would write it: This graph is vertically compressed by the factor of 0.5 (or any number less that 1).
- Vertical Reflection: If the 'a' is a negative number then it means the graph is vertically reflected over the x axis downwards.
- Translations: If there is a number after x squared like in y=a(x-h)²+5 (the 'h') then it means that the vertex has been translated right or left. The way you would write it: The vertex has been horizontally been translated 3 (or any other number) units to the right/left. Also if there is a 'k' then that means that the vertex has been translated up or down. The way you would write it: The graph has been vertically translated 4 (or any other number) units up/down.
Graphing Factored Form
Graphing Standard Form
Flight
To solve a flight word problem you need to know the quadratic formula and the axis of symmetry. Here is an example of a flight word problem:
A ball is thrown upwards at an initial velocity of 8.4m/s, from a height of 1.2 m above the ground. The height of the ball, in meters, above the ground after t seconds is modeled by the equation h=-4.9t²+8.4t +1.2.
1. How long will it take for the ball to fall to the ground, rounded to the nearest tenth of a second?
2.What is the maximum height of the ball? At what time will it reach this height? Round your answers to the nearest tenth.
Factor Quadratic Expressions of the Form ax² + bx + c
The flight of a ball is modeled by the equation ℎ = −5² + 20 + 25 where h represents the height of the ball in meters, and t represents the amount of time the ball has spent in the air in seconds.
a. Write the equation in factored form, determine the x-intercepts and vertex, and graph.
b. When does the ball hit the ground?
c. What is the highest the ball flies, and when does that happen?
X-Intercepts & Vertex
Graph
Using the X-Intercepts and Vertex to answer other questions
Optimization
You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?
Reflection
Multiple Choice
Short Answer
Factored Form & Vertex Form Question
Application 1
Application 2
Communication
How does Quadratics 1, 2 & 3 relate?
In quadratics 1, we mostly focused on solving vertex form by finding axis of symmetry, optimal value, zeros, and determining the step pattern. With all this information, we were able to graph it.
In quadratics 2, we focused on expanding and factoring. We learnt how to find axis of symmetry, optimal value, and zeros. We also did this in quadratics 1, but the only difference was that there might have been some different methods to do this.
In quadratics 3, we focused on rewriting standard form equations into vertex form by completing the square and by learning a new equation which is called the Quadratic Formula. We also learnt how to graph quadratics using the x-intercepts which we also learnt in quadratics 1.
In quadratics 1, 2 & 3, the word problems were mostly all asking for the same stuff just in different forms of equations and different methods of asking them. For example, what is the max height? They are asking for you to find the vertex and state the 'y'. In vertex form {y=a(x-h)²+k}, the vertex is (h,k) and so you already know your max height by just looking at the equation and knowing 'k' is your 'y-intercept'. In standard form {y=ax²+bx+c}, you know that the 'c' is the 'y-intercept' which is your max height. Now in factored form, you have to do some solving to find the max height, starting with finding the zeros, then the AOS, & lastly the optimal value. The AOS and optimal value would be your vertex, at which the optimal value is the 'y-intercept'.
As you can see, all of the parts of quadratics are related in several ways.