Modeling Relationships: Linear and Quadratic Functions

IB topic 2 - Chapter 3

What does a trinomial look like?

Ax² + bc +c

2 types of factoring trinomials - Think: what do you look for first?

1) when a = 1, you can just distribute factors of AC that are the sum of B

2) when a > 1, you need to split b (with the 2 factors) but still enter into the trinomial. Then look for the GCF from the new split equation (writhing the trinomial) - from their you should have 2 () that are the same - and create your second () out of the GCF’s you placed on the outside

Gradient

IB term for slope (still = m)

How to factor quadratics in () and () format (factored format)

You can use the FOIL Method,

Factoring with 2 perfect squares (the difference of)

You can use the sq root, as the 2 factors (and proceed with your factor formatting)

How do you factor with perfect squares trinomials

Find the sq root - of your 2 square terms (ideally a and b) - those are your factored terms (in factoring format) - then simply square your one factored terms (because the equations itself is already square)

What does the formula for Gradient look like? - this will be in your formula booklet

M=y2-y1/x2-x1

How do you find a parallel line?

If line one and line 2 are parallel then gradient of 1 and gradient of 2 will be the same

How do you tell if lines are perpendicular?

If line one and line 2 are perpendicular, then slope one * slope 2 will equal -1

What are the parameters of the Y = Mx + c equation:

M (gradient) and C (y intercept)-

What is the gradient of a horizontal line?

0

What is the gradient of a vertical line?

Undefined

Equation of a horizontal line:

(0,c), y = c

Equation of a vertical line:

(B,0), is x = b (think of the overall y= Mx + c)

What is gradient intercept form?

Y = mx + c

What is point gradient form? What does it tell you?

Y-y1 = m(x-x1), Parameters are, gradient (M) and point a point(x1, y1)- this will be on the line.

What does general form look like? What are the parameters?

Ax + by + d = 0, used when using integers (integers: not decimals or fractions)

How to use your GDC - finding intersections:

Use y= first, enter your 2 line equations. Then hit graph. From there, you want to hit trace, then 2nd and hit trace again (for calc), and select 5 - or intersect. Then follow the instructions in the calculator- labeling each line before hitting enter angina to guess the intersection.

What do you call a linear function that describes the relationship between 2 variables?

Linear model

What is a linear model see for:

To analyze and predict how the dependent variable will change, in response to to the independent variable (think basic tables and relationship issues) - but know you need to know how to represent these algebraically

What is a quadratic function?

F(x) = ax² + bx + c; a cannot be = to 0

What is the graph of a quadratic function called?

A parabola

What does it mean when a parabola is concave down?

It has a maximum point

When a parabola is concave up?

That means the parabola has a minimum.

What is a vertex?

The max or min point of a quadratic function.

What is the axis of symmetry?

The dashed line through the vertex of a parabola

What is the parent quadratic function (the bare minimum)

Y = x², (0,0) is vertex

What is the ___= for transformations of the parent graph

Y= g(x)

What does a reflection look like:

Y=-x², looks like a reflection over the x axis

What dose a stretch look like:

Vertical stretch is change with scale factor, looks like y=ax² - multiplied by # if that is below 0 its a compression- above its a stretch.

What does a horizontal translation look like

Remember what you expect is opposite, so decimals move left, and full numbers move right. Y = (x -h) ²

What does a vertical translation look like,

This is up and down, y = x² + k, up is #s larger than 0 and down is numbers smaller than 0

Alternate way that translations can be represented:

(HK) on top of each other… H is horizontal and k is vertical. This can represent simple translations.

What does y = -f(x) mean

Reflection across x axis

What does y=f(-x) mean

Reflection across the y axis

Think about how it may be different than vertical dialations- Horizontal dialations:

Y=f(qx)- compared to vertical - variable is insider prentasise

What is vertex form?

F(x)=a(x-h)² + k - a cannot equal 0 - (h,k) are the vertex coordinates- and h is the axis of symmetry

Quadratic equation for General form: What is it

F(x) = ax² + bx =c

Equation for the equation of the axis of symmetry

X= -b/2a

Coordinates of a vertex are:

(-b/2a, f(-b/2a))

Vertex form of a quadratic equation?

F(x) = a(x-h)²+ k

Vertex form - quadratic (X intercepts)

(P,0) (q,0)

Axis of symmetry - intercept form - quadratic

X=p+q/2

What is the intercept quadratic function form?

F(x)=a(x-p)(x-q), a does not = . P and Q are coordinates of the x intercept.

Finding the axis of symmetry - with quadratic factorized form

X= (p+q)/2

Vertex of the quadratic factorized form: - not on formula booklet

(P+q/2, f(p+q/2))

How to switch quadratic general form to a factorized form:

Use your a>1 and 1>a differences - stil the same - just know to add what x= at the end

Finding a quadratic function of a graph:

Start with a(x-p)(x-q) form - factor accordingly