Quadratic Functions

Quadratic Equations

Standard form

y=ax^2+bx+c

Parabola characteristics

Parabola

A U- shaped curve with certain specific properties.

Vertex

The point where the curve changes direction is called the vertex

A(x) value

The Axis of Symmetry

The types of roots

Two Distinct Roots

One Real Root

No Real Roots

Reflection in the X- axis

A reflection is the flipping of a point or figure over a line of reflection.

Vertical reflections in x-axis if a<0

A(y) value

Optimal value

Minimum Value

The Minimum value of a Quadratic function is the lowest point of the Parabola

Maximum value

Its the Highest peak of the Parabola

Y- Intercept

The y-intercept in any equation is the point at which the line crosses the y-intercept

y=x^2

Is the point (0,0) which is the vertex.

The A value

Direction of Opening

Either up or Down. If a > 0, then the parabola opens up , if a < 0, then the parabola opens down

Ways to solve Quadratic Equations

The X intercepts

The Quadratic Formula

Is a way to solve any quadratic equation

Are the points or point at which the parabola intersects the X- axis

Completing the square

Completing the square is a method of solving a quadratic equation by changing it's form.

1) Change the equation, so the constant term C, is by itself.

2) If (a), the leading coefficient is not equal to 1, divide both sides by A.

3) Add the square of half the coefficient of the X term to both of the sides

4) Factor the left side as the square of a binomial

5) Take the square root of both sides

6) Solve for X

The Foil Method

A very basic method of multiplying two binomials to find the equation.

Graphing

Relations

Linear Relations

A linear relation is a relation between two variables that appears as a straight line when graphed.

Non-Linear Relations

A non-linear relation is a relation between two variable that does not forma straight line when graphed.

Plotting Points

Scatter plot

Scatter plot is a graph where there is no revealing correlation.

Vertex Form

y= a(x-h)^2+k

Translation

Vertical Translation

Vertical translation moves the graph up or down

Horizontal Translation

Horizontal Translation moves the graph left or right

Transformation

Stretches and Compressions

If y=f(x), then y= af(x) gives a vertical stretch when a>1 and a vertical compression when 0<a<1

Factored form

y= a(x-r) (x-s)

Trinomial Factoring

Factoring Trinomials means finding two-binomials that when multiplied together produce the given trinomial.

5x^2-28x-12

=(5x+2) (x-6)

The zero product property

The Zero Product Property basically, states that a product of factors is zero only if one or more of the factors is zero