Everything You Need to know About Quadratics
Learning Goals for Quadratic Unit
- I could use a Table of Values to create a graph
- I could Identify the difference parts of the Quadratic Equation
- I can graph the base graph of a quadratic function with no technology
Introduction of the Quadratic Unit
What is a Parabola?
A Parabola is a curve that is at equal distance from the fixed point and the fixed line (Line of symmetry)
Overview of Vertex Form
The formula of a quadratic in a vertex form is:
y=a(x-h)2 + K (Note that the 2 is a squared)
The value of "A" in this formula, determines whether the parabola is stretched (Narrow) or is compress (Wider). As the value of "A" increase, the more narrow it gets. As value of "A" decreases, the parabola gets wider. It also tells you the the direction of opening of the parabola. If the "A" value is a negative, the direction of opening is down and vice versa.
The "H" in this equation determines whether the parabola will move to the right on the graph or left in the graph. This is more tricky than "A" and "K" because this is backwards. If the value of "H" is -2, the parabola moves right on the graph. But if it is a +2 in the equation, it moves to the left of the graph.
The "K" in this formula, determines whether the parabola will move up or down on the graph. If the "K" is 2, it moves 2 spaces up and if the value of "K" -3, the parabola moves 3 spaces down.
The value "H" ( X) and "K" (Y) are your vertex. But make sure you change the value of "H" to the opposite (-2 to 2).
How to Solve A Quadratic Word Problem?
The Maximum Height is the "Y" intercept of the vertex.
Time Reached Maximum Height is the "X" intercept of the vertex.
To find the Initial Height, you must substitute the "X" with zero. To find your answer.
To find the Height of the object at a specific time, you substitute the "X" with the specific time.
Word Problem (Vertex Form)
y= -(x-4)2 + 4. Where y equals height in meters and x equals time in seconds.
A) At what height did the ball reach the maximum height? and at what time?
y= - (x - 4)2 + 4 (x,y) to (h,k) to (4,4)
The h and k in this equation are the answer.
Therefore the maximum height in which the ball reached was 4 meters at 4 seconds.
- I could factor using different methods
- I need to know what a,r and s in the Factoring Equation are so that you know how to use it right
Overview of Factoring Form
The "A" in this equations tells you the direction of the opening of the parabola. If "A" is a negative in the equation, the direction of opening is downwards. Also, if "A" is a positive, the direction of opening, is upwards of the parabola. "A" also determines the shape of the parabola (compress or stretched)
The "R" and "S" in this equation are the x-intercepts of the equation.
Axis of Symmetry:
To find the A.O.S, you sub the x value in the equation to find the optimal value.
x= (r+s/2) is the equation you substitute the x value to find the optimal value.
To find the y-intercept, you set x as 0. With this you solve for the y-intercept.
Expanding and Simplifying (Factored Form)
(3x + 2)(2x - 5)
= 6x2 - 15x + 4x - 10
= 6x2 - 11x -10
Monomial Common Factor (GCF)
3x + 6y
= 3(x + 2y)
Binomial Factoring (GCF)
Factoring by Grouping
Factoring Simple and Complex Trinomials
Perfect Square Trinomials
Such as: 25x2 + 200x + 4
Differences of Squares
An example of Difference of Square is:
(3x - 8)(3x + 8)
Word Problem (Factored Form)
Let x rep smallest integer
Let x + 2 rep 2nd integer
Let x + 4 rep 3rd integer
x + x + 2 + x +4 = 231
3x + 6 = 231
3x = 231 - 6
3x = 225
3x = 225/2
x = 75
Therefore the three integers are 75, 77 and 79
Overview of Standard Form
y=ax2 + bx + c
The value of "A" gives you the shape and the direction of opening of the parabola. The value of "C" is the y-intercept of the parabola.
- I could graph the parabola using the Quadratic Formula
- I could find the x-intercepts using the Quadratic Formula
Completing the Squares
(ax2 + bx) + c
If the a has a value more than one you must divide it out of the equation to isolate the x.
Then you have to divide b by 2 and then square it, so you can add the number within the bracket.
You then subtract it in the bracket and then subtract it out of the bracket to balance the equation.
You then take the number you square it by and then substitute it into the vertex form equation.
Reyad is practicing her circus act by flying out a cannon. The equation that represents her flight is h= -5t2 + 20t + 2. With her height represents as h, in meters and t represents time in seconds.
A)What is the maximum height that Ms.Dhaliwal reaches?
B)At what time did she reach the maximum height?
The quadratic unit was a tricky unit. It had three different parts; vertex form, factored form and standard form. The equation I like to use the most is the standard form equation. The reason for this is because I find it easier to find the solution, because since the quadratic formula has pretty long, you find the solution with less effort.
Factoring connects to graphing by when you factor you find the expression for the x-intercepts, which you can use to find the x-intercepts of the parabola. This connects to graphing by you use factoring to find the vertex of the parabola, which you can use to plot on the graph.