Quadratics

What are the three forms of a Quadratic Equation?

Vertex Form: y = a(x - h)² + k
(h,k) —> vertex

Standard Form: y = ax² + bx + c

c —> y-int,

Intercept Form: y = a(x - p)(x - q)
p & q —> x-ints

How do you find the vertex from standard form of a quadratic (h,k)

y = ax² + bx + c —> you do: b/2a (h), then plug that value for x in the equation
a(h)² + b(h) +c —> that value is your k

Equation for axis of symmetry in standard form

-b/2a

What is the domain for all quadratics

ARN (All Real Numbers)

How do you determine if there was a reflection in a quadratic equation

By the a value (if its negative or positive)

How do you determine if there was a dilation in a quadratic equation?

By the a value (anything above 1 is a stretch, under 1 is a compression)

How do you find the y-intercept in a quadratic equation

Set the x’s in the equation to 0

-4(0)² + 16(0) -5

How do you determine the concavity of a quadratic (up or down)

The sign of the a

What is the relationship between h value in the vertex and the axis of symmetry 

They are the same value

How do you convert intercept form to vertex form

Calculate h value of vertex using midpoint formula for x-ints, then plug in the h value into the equation.

(5 + 3)/2 —> h value

3(4-5)(4-3) —> k value

How do you convert standard to vertex form

Calculate h, by using: b/2a, then plugging h into equation as x.

How do you convert vertex to standard form

Expand the equation and combine like terms, them multiply by a and add the k value

-2(x-3)² + 5
-2(x-3)(x-3) + 5
-2(x² - 6x + 9) + 5

-2x²+12x -18 +5

-2x²+12x - 13

How do you convert vertex to intercept

Convert vertex to standard then factor

How do you convert standard to intercept

Factor the equation to find the x-intercepts.

How do you convert intercept to standard

Expand the equation:

2(x + 3)(x - 1)

2(x² +2x -3)

2x² + 4x -6

How to determine the number of solutions? (real roots) (in standard)

Calculating the discriminant sqrt(b² - 4ac)

Positive = 2 solutions

Zero = 1 solution

Negative = no solutions

How to determine the number of solutions? (real roots) (in intercept form)

The opposite sign of the x-ints (p and q)

What is each form used for:

Standard: finding y-int,
Intercept: finding x-ints

Vertex: finding vertex

If the ball lands 60 ft from the kicker, how far from the kicker would you expect the ball tor each its maximum height (20 ft).

30ft away from kicker (halfway), (h,k) (30,20)