Unit 5: Quadratic Equations and Functions

Trinomial

A polynomial with three terms

Binomial

A polynomial with two terms

root

A solution to a quadratic equation

Linear Term

A term with the variable has a of degree one

Quadratic Term

A term with the variable has a of degree two

quadratic equation

An equation whose greatest exponent is 2

Solving by completing the square

Equation used in order to solve is b/2^2, "b" coming from the bx term

Quadratic Equation

Polynomial with degree 2

What are the zeros of a function?

Same as x- intercepts, roots, solutions to equations

A quadratic equation will have two solutions when...

The graph crosses the x axis in two places

maximum

The highest point of a parabola.

axis of symmetry

The line that cuts the graph into two symmetrical parts; x = -b/2a

minimum

The lowest point of a parabola

vertex

The point (x,y) of a parabola where it crosses the axis of symmetry.

Parabola

The shape of a the graph of a quadratic function

Constant Term

a term that has a number but no variable

domain of a quadratic function

all real numbers

Quadratic formula

allows to calculate the solutions of any quadratic equation.

discriminant

b^2-4ac

Discriminant

b²-4ac

Zero Product Property

if ab=0, then a=0 or b=0

solving by graphing

sometimes when graphing there will be two solutions, either positive or negative. There will also be only one solution which is zero, and there will also be a no solution problem.

Complete the Square

take half of b, then square it. Add to both sides of equation and factor down to vertex form.

A quadratic equation will have no real solutions when...

the parabola does not cross the x axis

y intercept

the point where the graph crosses the y axis

A quadratic equation will have one solution when...

the vertex of the parabola is on the x axis

range of a quadratic function

the y values

Vertex form

vertex= (h, k)

downwards parabola

when "a" is negative

upwards parabola

when "a" is positive

Solve (x + 10)(x + 2) = 0

x = -10, x = -2

Solve (x + 5)(x - 1) = 0

x = -5, x = 1

Solve (x - 10)(x - 2) = 0

x = 2, x = 10

Solve (x - 5)(x + 1) = 0

x = 5, x = -1

Where are zeros of the quadratic equation located on the graph?

x intercepts

Standard form

y intercept: (0, c)

Factor x^2 + 20x + 100

(x + 10)(x + 10)

Factor x^2 + 10x + 25

(x + 5)(x + 5)

Factor x^2 - 20x + 100

(x - 10)(x - 10)

Factor x^2 - 10x + 25

(x - 5)(x - 5)

factored form

(x+3)(x-7)

vertex form

(x-3)^2 - 8 = 0

How many solutions can a quadratic equation have?

0, 1 or 2

To make x^2 + 10x + _____ a perfect square trinomial, add

25

standard form

3x^2 +7x+1 = 0

To make x^2 + 50x + _____ a perfect square trinomial, add

625

To make x^2 -6x + _____ a perfect square trinomial, add

9

Standard Form

This quadratic function is written in ...

Vertex Form

This quadratic function is written in....

solving by square roots

Write the original equation. Isolate x^2 on one side of the equation. Find the square roots of each side and simplify.