Algebra II Jumpstart Notes



Ch. 3 : Alg2-SOL Functions-Part 2

SOL A2.6, A2.7a-f, A2.8

AII.7-Functions

Be able to recognize the graphs for the following functions: linear, quadratic, absolute value, polynomial (cube and cube root especially), exponential, and logarithm functions.

Equation examples: [pic](Linear (degree of 1), [pic]( Quadratic (degree of 2), [pic](Absolute Value,[pic](Cube function, [pic](Cube Root, [pic](Exponential (a number raised to the x power), [pic](Logarithm

|Function |Equation etc |

|Linear |y = mx + b (SI) |

|[pic] |ax + by = c (SF) |

| |[pic] slope |

| |b is (0,b) y-intercept |

| |[pic] point on the line |

| | |

| |Horizontal line HOY |

| |y = #, zero slope |

| | |

| |Vertical lines are not functions (VUX) |

| | |

| |x = # undefined slope |

|Quadratic “U” Parabola | y = a(x – h)2 + k |

|Find Vertex by using “Calc”|Vertex (h,k) |

|key maximum or minimum |opposite same |

|[pic] | |

| |a>0 opens up, a< 0 opens down |

| | |

| |[pic]>1 stretch, [pic]0 opens up, a< 0 opens down |

| |[pic]>1 stretch, [pic]0 opens up, |

| |a< 0 reflects down |

| |[pic]>1 stretch, [pic]0 as graph on left , a< 0 reflects |

| |[pic]>1 stretch, [pic] 1 |

|[pic] | |

| |y = k is the horizontal asymptote |

| |e [pic][pic] 2.72 (natural log base e)|

|Logarithmic |[pic] (inverse of exponential). |

|[pic] |[pic] = [pic] |

| |x = h is vertical asympt. |

| |Log is log base 10 |

| |Ln is log base e |

Polynomials: To find the zeros of a polynomial equation, either:

1.) Graph the equation on your calculator and look at where the graph crosses/touches the x-axis

or

2.) Solve the equation by factoring and setting each factor = 0 (may need to use the quadratic formula (given to you on the ‘formula’ screen.) You must do this when you cannot tell where the graph crosses or if it doesn’t cross the x-axis.)

|Polynomials |Zeros |Types |Turns |End |

|Example: Cubic | | | |Behavior |

|Degree 3 |1. Real Zeros are the|1. If there are no x |1. The maximum | |

|[pic] |x values of the x |ntercepts there are |number of turns is |1. If the leading coefficient|

| |intercepts. |no real zeros, (all |equal to the |(LC) is ‘+’ the right |

| | |zeros will be |degree – 1. |behavior rises, if the LC is |

| |2. Zeros are also |imaginary) | |‘-‘ the right behavior falls |

| |called roots, or | | | |

| |solutions |2. A tangent implies | |2. If the degree is even, |

| | |a double root | |right and left behavior will |

| |3. If the zero is x |(repeated solution) | |be the same, if the degree is|

| |= h, then its factor | | |odd right and left behavior |

| |is (x-h) |3. Irrational zeros | |is opposite. |

| | |come in pairs as do | | |

| |4. The number of |imaginary zeros | | |

| |zeros = the degree | | | |

| |(this includes real, | | | |

| |imaginary and double | | | |

| |roots) | | | |

Finding Domain/Range,

A ‘Function’ means that x-values do not repeat---it must pass the vertical line test.

Domain – set of all x-values Range – set of all y-values

Ex 1: Find the Domain/Range of [pic].

From the graph shown: (Note: [pic] symbol means “all reals”)

Domain =

Range =

Increasing/Decreasing Intervals

As x increases from - infinity to + infinity (read from left (right), do y values increase or decrease? The intervals will be the x values in these areas.

Ex 2:. From the graph shown, one increasing interval would be from – infinity to 0. What is the other increasing interval?

Ex 3: What are the decreasing intervals?

Leading Coefficient

Ex 5. What is the sign of the leading coefficient for the graph above? _____________

Polynomial practice: Answer the examples below:

Ex 7: What is the greatest number of real zeros possible for[pic]? _____

Ex 8: What are the zeros and corresponding factors of the graph given below?

Example 9: What is a possible equation for this graph?

Transformations of Graphs:

Ex. 10: Give the name of the graph and identify the transformations from the parent:

Ex. 8: What are the possible equations of these graphs?

Rational Functions: See the chart for information on rational graphs:

|Rational Function |y = [pic] where p(x) and| Domain all real numbers|Vertical Asymptotes: Set q(x) = 0 and solve.|

| |q(x) are polynomial |except the values that |Look at domain restrictions. |

|[pic] |functions |make q(x)= 0 | |

| | | |Horizontal Asymptotes: |

| |q(x) [pic]0 |Zeros of function set |1. Degree of p(x) < Degree of q(x) y = 0 |

| | |p(x)= 0 and solve |2. Degree of p(x) > Degree of q(x) None |

| |discontinuous | |3. Degree of p(x) = Degree of q(x) |

| | | |y = LC of p(x)/LC of Q(x) [pic] |

Domain/Range of Rational Functions: Will depend on the asymptotes of the graph.

Ex 2: Find the Domain/Range of [pic].

Vertical asymptote? x = ____ Horizontal asymptote? y = _____

Graph on calculator to help find the Domain/Range:

Domain: Range:

Answer the questions below each graph:

9. 10. 11.

[pic] [pic] [pic]

Name: _______________________

Degree: _____________________

LC: + or -

# of Zeros: _______________________

Real Zeros: _____________________

Double roots? ______________

Factors: _________________________

Domain _____________________

Range ______________________

y-intercept ________________

Increasing? ____________________

12. 13. 14.

[pic] [pic] [pic]

Name: _______________________

Domain _____________________

Range ______________________

Horizontal Asymptote: _________

Vertical Asymptote: ____________

-----------------------

Name: _______________________

Degree: _____________________

LC: + or -

# of Zeros: _______________________

Real Zeros: _____________________

Double roots? ______________

Factors: _________________________

Domain _____________________

Range ______________________

y-intercept ________________

Increasing? ___________________

Name: _______________________

Degree: _____________________

LC: + or -

# of Zeros: _______________________

Real Zeros: _____________________

Double roots? ______________

Factors: _________________________

Domain _____________________

Range ______________________

y-intercept ________________

Decreasing? ___________________

Name: _______________________

Domain _____________________

Range ______________________

Horizontal Asymptote: _________

Vertical Asymptote: ____________

Name: _______________________

Domain _____________________

Range ______________________

Horizontal Asymptote: _________

Vertical Asymptote: ____________

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