INTRO TO QUADRATICS
Solving Quadratic Equations
By Factoring
By Completeing the Square
Example:
-5x²+20x-2=0
1. Block off the first two terms→ (-5x²+20x)-2=0
2. Factor out the A→ -5(x²-4x)-2=0
3. Add "zero" by: dividing the middle term by 2, and squaring it→ -5(x²-4x+4-4)-2=0
4. Bring out the negative→-5(x²-4x+4)+20-2=0
5. Factor inside the brackets and simplify→-5(x-2)²+18=0
6. Divide equation by the number in front of the brackets to get rid of it→(x-2)²-18/5=0
7. Bring the second term over to the other side, and square root all to get rid of the square on the first term→√(x-2)²=√18/5
8. Isolate x, and simplify→x=±3.6+2
9. You will get two answers because the number that was square rooted is positive and negative→ x=5.6 or x=-1.6
By Using the Quadratic Formula
Properties of Quadratic Relations
Important Terms and Characteristics
Vertex→the point on the graph with the greatest y-coordinate (if the graph opens down) or the least y-coordinate (if the graph opens up); vertex is directly above or below the midpoint of the x-intercepts (if the parabola crosses the x-axis)
Optimum Value→the y-coordinate of the vertex corresponds to the optimum value, either a maximum or minimum value
Minimum Value→ y-coordinate of the vertex when the graph opens up
Maximum Value→ y-coordinate of the vertex when the graph opens down
Direction of Opening→ if the constant value of the second differences is positive, the parabola opens up; and if its is negative the parabola opens down
Axis of Symmetry→ a vertical line that passes through the vertex (perpendicular bisector) that makes the parabola symmetrical
Equation of the Axis of Symmetry→ As the coordinates of the vertex are (h,k), the equation would be x=h since the line passes through the vertex
Zeros/x-intercepts→ the x-coordinates of the points the parabola crosses on the x-axis
Graphing Factorable Quadratic Equations
Three Things to Consider When Graphing
2. y-intercept: set x=0, and determine this point
3. vertex: the maximum or minimum point of the parabola; very important when solving optimization questions
The video below will demonstrate how to find these three things when your equation is factored, and use them to graph your quadratic equation.
Stretching, Reflecting and Translations of Quadratics
Parent/Original/Base Parabola
Stretching
- parabola gets thinner
- y-values increase
- if the A value is greater than 1
Vertical Compression
- parabola becomes wider
- y-values decrease
- if the A value is less than 0 and greater than 1 or less than -1 and less than 0 (fractions)
Reflecting
Translations
In y=x²+k:
- the "k" produces a vertical translation up or down
- the parabola does not stretch or compress (as there is no "a" value)
- affects the y-values
In y=(x-h)²:
- the "h" horizontally translates the parabola left or right
- affects the x-values
Different Forms of Quadratic Equations
Standard Form
A value:
-if the parabola opens up or down
-vertical stretch or compression
C value:
-y-intercept
Factored Form
A value:
-opens up or down
-vertical stretch or compression
R and A values:
-x-intercepts
Vertex Form
y=a(x-h)²+k
A value:
-opens up or down
-vertical stretch or compression
H value:
-x-coordinate of vertex
-horizontal shift
-axis of symmetry
K value:
- y-coordinate of vertex
- vertical shift
- optimum value