Vertex Form
Amour Ahmed
LEARNING GOALS
- We can use the 1st and 2nd differences from a data table to determine if a equation is linear, quadratic or neither.
- We can use a table of values to create a parabola
- We can graph a basic quadratic equation without using technology
An Intro To Quadratics
Different Directions of Parabola
Parts Of A Parabola
- Vertex
- Y- Intercept
- X-Intercept
- Axis Of Symmetry
What Is A Parabola ?
A parabola is the graph of a quadratic equation/function. It an be either in the shape of the letter 'U' or it can be in the shape of the inverted letter of 'U'.
Key Features Of Quadratic Relations
What Do We Know About Parabolas ?
- A parabola can either open up or down.
- The zeros of a parabola is where it crosses the x-axis
- The Axis of symmetry divides the parabola into two equal parts
- The vertex of a parabola is the point where the AOS and the parabola meet.
- The vertex is a y-co-ordinate which is optimal value
- The y-intercept of a parabola is where the graph crosses the y-axis
- The optimal value is the value of the y co-ordinate of the vertex
- The vertex is the point where the parabola is at its maximum or minimum value.
Different Forms and Types of Quadratic Equations
A quadratic equation is any equation having the form ax2 + bx + c = 0. This is where a, b and c are known values. a can't be 0."x" is the variable or unknown (we don't know it yet). If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and are known as the quadratic coefficient, the linear coefficient and the constant.
First Differences
First differences are the changes in y-values for consecutive x-values in a table. On this table the first differences are constant by - 3, this shows that the graph will be Linear.
Second Differences
Second differences are changes in the consecutive first differences. On this table the first differences are not constant so we move on to second differences to determine if its Quadratic.
How Can You Tell If A Line Is Linear or Non-Linear (Quadratic) ?
Well....
Linear Relationships: Points that follow a straight line are known as linear relationships.
Quadratic Relationship: If points follow a curved line (parabola), the relation is quadratic.
Here are examples you can try to guess...
What Can You Tell From A Vertex Form Equation ?
Vertex form gives us the value of the axis of symmetry and the optimal value. The h values gives the axis of symmetry. The value of k gives away the optimal value. Vertex form also gives away the stretch with the factor that represents a. If the value of a is greater than 1, the parabola is vertically stretched. If the value of a is less then 1, then the parabola is horizontally compressed.
Example
Formula
Something Called The Mapping Formula ?!
The general mapping rule for quadratics is
(x,y) → (x + h, ay + k)
The variables a, h and k can be found in the turning point formula: y = a(x - h)² + k
Example: y=4x2+16
To get y = 4x² + 16 to turning point formula, you must complete the square
y = 4x² + 16
Divide each side by 4 to get rid of any factors before x²
y/4 = x² + 4
y/4 = (x + 0)² - 0 + 4
y = 4(x + 0)² + 16
Therefore: a = 4, h = 0, k = 16
Substitute these values into (x,y) → (x + h, ay + k)
(x,y) → (x + 0, 4y + 16)
Therefore: (x,y) → (x, 4y + 16)
How To Write Quadratic Equations Given The Vertex And A Point
Just as quadratic equations can map a parobla, the parobla points can help write a corresponding quadratic equation. In vertex form, y=a(x-h)^2+k, the variables h and k are coordinats for the parobla's vertex. The x point sign much change though.
First substitute the vertex's coordinates for h and k in the vertex form equation. For example let use a vertex of (2,3). The equation would be y=a(x-2)^2+3.
Then substitute the point's coordinate for x and y in th equation. In this case w can use points (3,8). The new equation would be 8=a(3-2)^2+3.
After solve the equation for a. In this example solving for a would result in a becoming 5.
Lastly subsitute the values of a into the first equation. The final answer should be y=5(x-2)^2+3.