Grade 10 Academic Math
Table of contents
-What is a parabola?
-Analyzing a parabola (second differences)
Types of equations
- Understanding the step pattern
- Describing Transformation
- Graphing vertex form x
- Isolating for a variable
-Finding optimal value
-Finding axis of symmetry.
- Factors and zeroes
- Finding axis of symettry
- Finding optimal value
- Factoring using algebra tiles
- Expanding factored form equations to standard form
- Factoring simple trinomials
- Factoring complex trinomials
- Other forms of factoring (differences of squares, perfect squares)
- Reviewing graphing with factored form
- Using quadratic formula
- Completing the square
What is a Parabola?
Properties of a Parabola
How to find out if an equation is quadratic
Extra Practice on calculating second differences
Breaking down and understanding the equation & transformations
-The Y variable represents the Y-intercept
-The A variable represents whether the parabola opens up or down (positive a value results to parabola opening down, negative a value results to the parabola opening down), it also represents if the parabola will have a compression or stretch
- The X variable represents the x intercept
- The H variable represents where the X point is on the graph (right or left) (axis of symmetry) (If h is positive 6 the parabola will be horizontally translated 6 units to left)
- The ^2 creates the arch shape of the parabola
- The X variable is in charge of whether the parabola will be vertically shifted up or down (according to positive or negative sign) (positive K value will shift parabola up and negative K value will shift parabola down)
After you plot your vertex using the transformations explained you can use the step pattern to plot your vertex form equation!
With all variables at 0 the parabola would be at the origin at 0,0
- The (a) value stretches or compresses the parabola. If the (a) value is greater than 1 the parabola will stretch by the factor of the value of the variable; This is what is called a vertical stretch. If the (a) value is less than 1 the parabola will compress by the factor of the value of the variable; this is known as a vertical compression.
- The (-h) represents whether the parabola moves left or right. *RECORD (When the h value is positive the parabola moves left and when the h value is negative the parabola moves right). This is called a horizontal translation.
- The (k) variable represents whether the parabola moves up or down. *RECORD (Whenthe k value is positive the parabola moves up, when the k value is negative the parabola moves down). This is called a vertical translation.
Did you know? The H value and K value make up the vertex?
The Step Pattern
Over 1 up 1
Over 2 up 4
Over 3 up 9
Note: *When using the step pattern we just square the number we are going over by to find the number we go up by
Note: *When there is a whole positive/negative: number, fraction, decimal in place of variable a value you just simply multiply the a value to the squared part of the step pattern
finding Axis of symmetry in vertex form
In the equation the value of h is +5, so when the sign is flipped the axis of symmetry is -5.
Finding optimal value of a standard form equation.
Therefore the k value is positive 6, meaning the optimal value of the parabola will be +6 over the x-axis.
Isolating for x in vertex form (finding x intercepts/roots)
Graphing vertex form
Finding Zeroes in factored form
Finding axis of symettry in factored form
For example: X int @(6,0), (-4,0)
1=Axis of symmetry Therefore after averaging the x intercepts 1 is the axis of symmetry.
Finding the optimal value for equations in factored form
Factoring simple trinomials
Factoring Complex Trinomials
Special cases of factoring (difference of squares and perfect squares)
Using algebra tiles to factor (particularly for visual learners)
Expanding Factored Form equations to standard form
video about Graphing with factored form (made by myself)
The Quadratic formula
Quadratic Formula Song
Finding x intercepts of standard form equations using the quadratic formula
Finding the axis of symmetry of a standard form equation
Lets do an example, let's find the optimal value for the equation we found x-intercepts for above (3x^2-4x+1).
Method 1 (adding both x intercepts and divide by 2)
Axis of symmetry: (1+1/3)/2
Method 2- Using -b/2a (variables were identified in previous part)
Axis of symmetry: -b/2a
Therefore we applied both methods taught to find for the x intercepts of a standard form equation and we managed to get the same answer proving that our answer is correct.
Finding the optimal value of a standard form equation
With the example used above for explaining how to calculate x intercepts and axis of symmetry, we will also use the same example to explain optimal value.
Equation being subbed into: (3x^2-4x+1=0 X=0.66 (Axis of symmetry)
Therefore you have your optimal value.
- negative there is no solution
- is equal to 0 there is only 1 solution
- if greater than 1 there is 2 solutions.
Completing the Square
Optimization Fencing problem
Below is a picture of my factoring unit test: