chapters 2 - 7

Chapter 2

2-1 Relations and Functions:

* a function rule is an equation that represents an output value in terms of an input value. You can write a function rule in the function notion

ex

[Outputs: y, f(x), C, &, etc.

[Inputs: x, or any other variables

Flashcard 9

^^EX. Tickets to a concert are available online for $35 each plus a heading fee of $2.50. The total cost is a function of the number of tickets bought. What function rule models the cost of the concert tickets? Evaluate the function for 4 tickets.^^

Answer:

Independent = # of tickets bought (x)

Dependent = Total cost of everything (y)

2-2 Direct Variation:

You can write a formula for a direct variation function as y=kx, or y/x = k, where k = 0.

x represents input values, and y represents output values. The Formula y/x says that, except for (0,00, the radio of all output-input pairs equals the constant k, the constant of variation.

In direct variation, y/x is the same for all pairs of data where x = 0. So [y1/x1 = y2/x2] is true for the ordered pairs (x1, y1) and (x2, y2), where neither x1 nor x2 is zero

2-3 Linear function and Slope - Intercept Form:

[Slope - is also denoted with the letter m.

The slope is the “rate of change”.]

a function whose graph is a line a linear function. You can represent a linear function with a linear equation, such as y=6x -4. A solution of a linear equation is any ordered pair (x,y) that makes an equation true.

There are different special forms of the linear equation: slope-intercept form and point-slope forms.

2-4 :

Slope intercept form: use this form when you know the slope and the y-intercept.

Point - Slope form y-y1: m(x-x1): use this form when you know the slope and a point, or when you know two points.

Standard Form: A, B, and C are real numbers. A and B cannot both be zero

2-6 Families of Functions

• for a positive constant k and a parent function (fx), f (x) positive or negative k is a vertical translation for a positive constant h f (x positive or negative h) is a horizontal translation

• The linear functions form a family of functions, each linear function is a transformation of the function y= x. function y = x is the parent linear function.

• For a function f(x), the graph in the x-axis, the y- values change the sign and the x-values stay the same

  • concept summary translations of f(x)

  vertical translations

  • translation up k units, k>0

    y = f(x) + k

  • translation down by k units, k>0

    y = f(x) - k

• \

  • Vertical stretches and compressions

  • vertical stretch, a >1

    y = af(x)

  • vertical compressions, 0 <a<1

    y = af(x)

  • horizontal translations

  • Translation right h units, h>0

    y = f(x-h)

  • translation left h units, h>0

    y=f (x +h)

• \

  • reflections

  • in the x axis

    y = -f(x)

  • in the y-axis

    y=f(-x)

2-7 Absolute Value Functions and Graphs * need to work on Chapter 2

The absolute value of f(x), |f(x)|, gives the distance from the line y = 0 for each value of f(x)

When graphing especially the equation f(x) = |x| you will be able to notice that it is symmetric about the y-axis, which is the vertical line known as the axis of symmetry. this type of function has. A single maximum or a single minimum point is called a vertex.

ex. Do transformations of the form y = |x| + 2

answer: no, transformations of this form move the vertex up/down along the y-axis of symmetry, but the axis stays the same.

Key concept | general form of the absolute value function |

y = a | x+h| + k

The stretch or compression factor is |a|, the vertex is located at ( h,k), and the axis of symmetry is the line x=h

To determine which half-plane to shade, pick a test point that is the boundary. check whether that point satisfies the inequality. If it does, shade the half-plane that includes the test point. If not, shade the other half-plane. The origin, (0,0), is usually an easy test point as long as it is not on the boundary

+-

Chapter 3

solutions

• parallel lines indicate no solution
• intersection lines indicate one solution - the point of intersection
• coinciding lines indicate infinitely many solutions - all points on the line

use additive inverses to eliminate one of the variables. then solve for the other variable, the substitution to find the solution.

Solving systems with 3 variables

The graph of a linear equation with three variables is a plane. there a graph of a solution of a system of three linear equations in three variables will be

-planes intersecting in one common point ((one solution)

-pleases intersecting along a common ome (infinitely many solutions)

-no point lying in all three planes (no solution)

3-1 solving systems using tables and graphs

Concept Summary | Graphical Solution of Linear Systems

-intersecting lines: one solution, consistent, independent

-Coinciding lines: infinitely many solutions, consistent, dependent

-parallel lines: no solution, inconsistent

3-2 Solving systems algebraically

1. solve one equation for one of the variables (isolate it)
2. Substitute for that variable in the other equation
3. solve for the other variable

of the chapters

a function notion is

f(input) = Output

Ex. Function Notation

Functions are used in everyday life. For example:

• The cost of a good or service is a function of the demand
• An employee’s salary is a function of the number of hours they work.

By convention, we usually use the letter f to represent a function. However, if we are using a function for a specific application, it isn't uncommon to use a related letter. For instance, a function for the salary of an employee might be

s(hours)=dollars

or, written more simply,

s(h)=d.

Writing the full input for a function takes up a lot of space. Instead of writing “number” we can use a variable, like �.x.

So, we can represent

�(number)=number +1f(number)=number +1

as

�(�)=�+1f(x)=x+1

Writing the full input for a function takes up a lot of space. Instead of writing “number” we can use a variable, like �.x.

So, we can represent

�(number)=number +1f(number)=number +1

as

�(�)=�+1f(x)=x+1

The possible inputs for a function are called its domain.

We can specify a domain with an interval, and we can visualize it as a shadow on the �x-axis.

ertical shrink

When a function �(�)f(x) is multiplied by a factor 0<�<1,0<a<1, its points shrink toward the �x axis.

Show explanation

A quadratic either has a minimum or maximum point depending on which direction it's facing. This point is called the vertex of the function. The vertex is also a turning point — the function goes from increasing to decreasing or vice versa.