Unit 1 Objectives

Question 1

Find f(x+h) for any function f(x)

Answer 1

Plug in (x+h) for x and simplify.

Question 2

2. Find [f(x+h) – f(x)] / h for any function f (x)

Answer 2

The Difference Quotient

F(x+h)= plug in (x+h) for x

Subtract f(x)

Put all that over h

Simplify

Question 3

Find the domain and range of a function from an equation or a graph

Answer 3

Start with the domain being all real numbers, except what makes the function undefined:

1. Division by zero
2. The square root of a negative
3. Log of a negative or zero

Question 4

Find the combinations f + g, f – g, fg, and f/g and state their domains

Answer 4

Take two functions and add, subtract, multiply or divide them. Then simplify.

Question 5

Determine if a function is even, odd or neither and explain why

Answer 5

A function is even when f(-x)=f(x)
Which means the graph has symmetry about the y-axis.

It’s odd when f(-x)=-f(x)
Which means the graph is symmetric about the origin.

Question 6

Determine intervals over which a function is increasing, decreasing or constant.

Answer 6

Going up, down or staying the same is based on the y values, however it is described with intervals of x values. Similar to domain.

Question 7

Graph a function and find the maximum and minimum values. What are the terms associated with maximums and minimums?

Answer 7

Absolute maximum
Relative maximum
Relative minimum
Absolute minimum *a parabola with a negative coefficient has no absolute min.

Global maximum
Local maximum
Local minimum
Global minimum

Question 8

Graph and analyze all 12 of the functions in the Library of Functions. Name the 12 functions and their equation:

Answer 8

Constant f(x)=c

Linear f(x)=x or x^1

Quadratic f(x)=x^2 
Standard form f(x)=ax^2+bx+c

Cubic f(x)=x^3 or f(x)=ax^3+bx^2+cx+d

Quartic f(x)=x^4

Radical f(x)=x^1/2 or the “square root of x”

Cube Root f(x)=x^1/3 or the “cube root of x”

Exponential f(x)=a^x

Logarithmic f(x)=logb(x)

Rational f(x)=1/x

Absolute Value f(x)=|x|

Step Function f(x)=int(x)

Question 9

Graph piece-wise functions.

Name the steps to create a graph.

Answer 9

Make x/y t-charts for each equation of x within the given parameters.

Chart each point

Pay attention to included and not included values

Place arrows where they’re appropriate

Question 10

Standard form of quadratic equations:

Answer 10

f(x) = ax^2 + bx + c

Question 11

Vertex form of quadratic equations:

Answer 11

f(x) = a(x – h)^2 + k

Question 12

Find the vertex, x-intercepts, y-intercept, axis of symmetry and graph a quadratic function in standard form.

Answer 12

Vertex: -b/2a = x. F(-b/2a) = y

X-intercepts: set y=0 and solve for x

Y-intercepts: set x=0 and solve. In standard form, c equals the y intercept.

Axis of symmetry: x=-b/2a

Question 13

Find the points of intersection for two functions.

Answer 13

Set the functions equal to each other and solve

Question 14

Find the vertex, x-intercepts, y-intercept, axis of symmetry and graph a quadratic function in vertex form.

Answer 14

Vertex: is at (h, k).
F(x)=3(x-2)^2+1 so vertex = (2,1)

X-intercept: set y=0 and solve for x

Y-intercept: set x=0 and solve for y

Axis of symmetry: x=h
F(x)=3(x-2)^2+1 axis x=2

Question 15

Quadratic Formula

Answer 15

X = -b +|- the square root of b^2-4ac

All over 2a

Question 16

What is the discriminant?

What does it mean?

Answer 16

Discriminant is (b^2-4ac)

If >0: two real roots exist

If =0: one real root exists
(The vertex is on the x-axis)

If <0: it never crosses the x-axis
(Two imaginary roots exist)