Horizontal and Vertical Lines

Vertical lines:

Equation is x = #

Slope is undefined

Horizontal lines:

Equation is y = #

Slope is zero

Answer:4

21

4

1 xy

Answer: y = -4x - 16

Identify Relations that are FunctionsDetermine whether x = 3y2 represents y as a function of x.

x = 3y2 Original equation

Divide each side by 3.

Take the square root of each side.

First solve for y

Answer: No; there is more than one y-value for an x –value.

For each function, evaluate ƒ(-2) and ƒ(x+2).

ƒ(x) = x2 – 4x

f(x + 2) = (x + 2)2 – 4(x + 2) = = x2 + 4x + 4 – 4x – 8 = x2 - 4

2x + 1 if x ≤ 2

x2 – 4 if x > 2

h(x) =

Because –1 ≤ 2, use the rule for x ≤ 2.

Because 4 > 2, use the rule for x > 2.

h(–1) = 2(–1) + 1 = –1

h(4) = 42 – 4 = 12

Evaluate each piecewise function for x = –1 and x = 4.

Find Domains Algebraically

State the domain of the function .

Because the square root of a negative number cannot be

real, 4x – 1 ≥ 0. Therefore, the domain of g(x) is all real

numbers x such that x ≥ , or .

Set the bottom = 0

The domain is the values for which q(x) = 0 also.

Find the domain

x – 2 x2

– 1f(x) =

x – 2

(x – 1)(x + 1) f(x) =

Domain is all real except x = 1, x = –1 or (-∞, -1)U(-1,1)U(1,∞)

Find the average rate of change of f (x) = –2x 2 + 4x + 6 on the interval [–3, –1].

Use the Slope Formula to find the average rate of change of f on the interval [–3, –1].

Are either one of these functions?

Answer to a: Domain:{-4,-3,-1,2,3,5} and Range:{-3,0,1,2}Answer to b: Domain all real and Range: y≥ -1

Analyze Increasing and Decreasing BehaviorUse the graph of the function f (x) = –x

3 + x to estimate intervals on which the function is increasing, decreasing, or constant.

Answer: f (x) is decreasing on and

and increasing on

Graph the function f (x) = x 2 – 4x + 4 using a graphing calculator. Analyze the

graph to determine whether the function is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function.

Identify Even and Odd Functions

It appears that the graph of the function is neither symmetric with respect to the y-axis or to the origin. Test this conjecture.

f (-x) = (-x) 2 – 4(-x) + 4 Substitute -x for x.

= x 2 + 4x + 4 Simplify.

Since –f (x) = -x 2 + 4x - 4, the function is neither

even nor odd because f (-x) ≠ f (x) or –f (x).

Graph the function f (x) = x 2 – 4 using a graphing calculator. Analyze the graph to

determine whether the function is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function.

Identify Even and Odd Functions

From the graph, it appears that the function is symmetric with respect to the y-axis. Test this conjecture algebraically.

f (-x) = (-x)2 – 4 Substitute -x for x.

= x 2 - 4 Simplify.

= f (x) Original function f (x) = x 2 – 4

The function is even because f (-x) = f (x).

Answer: even; symmetric with respect to the y-axis

Graphing Piecewise Functions

g(x) =

1 4

Graph each function.

x + 3 if x < 0

–2x + 3 if x ≥ 0

Example Continued

O●

Once a hole is closed leave it closed.

The parent function f(x) = x2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g.

Use the description to write the quadratic function in vertex form.

Writing Transformed Quadratic Functions

4

3

g(x) = (x + 2)2 – 5 4

3

Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.

Adding and Subtracting Functions

(f + g)(x) = f(x) + g(x)

= (4x2 + 3x – 1) + (6x + 2)

= 4x2 + 9x + 1

Multiplying and Dividing Functions

= (6x2 – x – 12) (2x – 3)

Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function.

(fg)(x) = f(x) ● g(x)

= 6x2 (2x – 3) – x(2x – 3) – 12(2x – 3)

= 12x3 – 18x2 – 2x2 + 3x – 24x + 36

= 12x3 – 20x2 – 21x + 36

Set up the division as a rational expression.

Divide out common factors.

Simplify.

( )(x)fg

f(x)

g(x)=

6x2 – x –12

2x – 3=

Factor completely. Note that x ≠ .3

2

=(2x – 3)(3x + 4)

2x – 3

=(2x – 3)(3x +4)

(2x – 3)

= 3x + 4, where x ≠ 32

Multiplying and Dividing Functions

Evaluating Composite Functions

Step 1 Find g(4)

Given f(x) = 2x and g(x) = 7 – x, find each value.

f(g(4))

g(4) = 7 – 4

Step 2 Find f(3)

= 3

f(3) = 23

= 8

So f(g(4)) = 8.

g(f(x)) = g(x2 – 1)

Writing Composite Functions

= x2 – 1

2 – x2

(x2 – 1)

1 – (x2 – 1)

=

Given f(x) = x2 – 1 and g(x) = , write each composite function. x

1 – x

Writing and Graphing Inverse Functions

Switch x and y.

Solve for y.

Set y = f(x) and graph f.

f(x) = – x – 5. Then write the inverse and graph.

12

12y =– x – 5

12

x = – y – 5

x + 5 = – y 12

–2x – 10 = y

Write in y = format.y = –2(x + 5)

Juan buys a CD online for 20% off the list price. He has to pay $2.50 for shipping. The total charge is $13.70. What is the list price of the CD?

c = 0.80L + 2.50

In a real-world situation, don’t switch the variables, because they are named for specific quantities.

Remember!

c – 2.50 = 0.80L

c – 2.50 = L 0.80

Substitute 13.70 for c.

Evaluate the inverse function for c = $13.70.

The list price of the CD is $14.

L = 13.70 – 2.50 0.80

Check c = 0.80L + 2.50

= 11.20 + 2.50 = 13.70

Substitute.

= 14

Example Continued

= 0.80(14) + 2.50