Chapter 3 Review Pre-Calculus

Chapter 3 Review

Pre-Calculus

Determine what each graph is symmetric with respect to

y-axis y-axis, x-axis, origin, y = x, and y = -x

y-axis, x-axis, and origin

The graph of each equation is symmetric with respect to what?

Two squared terms, with same coefficients means it is an circle with center (0, 0)

Symmetric with respect to x-axis, y-axis, origin, y = x, and y = -x

Two squared terms, but different coefficients means it is an ellipse with center (0, 0)

Symmetric with respect to x-axis, y-axis, and origin

One squared term means it is a parabola shifted up 5 units and more narrow.

Symmetric with respect to the y-axis

Graph each equation:

Graph each equation:

Determine whether each function is even, odd or neither.

If all the signs are opposite, then the function is EVEN

Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither.

If all the signs are opposite and the same, then the function is NEITHER even or odd.

Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither.

If all the signs are the same,then it is ODD

Figure out f(-x) and –f(x)

Describe the transformation that relates the graph of to the parent graph

THREE UNITS TO THE LEFT

Describe the transformation that relates the graph of to the parent graph

THREE UNITS UP, AND MORE NARROW

Describe the transformation that relates the graph of to the parent graph

FOUR UNITS TO THE RIGHT, AND THREE UNITS UP

Describe the transformations that has taken place in each family graph.

Right 5 units

Up 3 units

More Narrow

More Narrow, and left 2 units

Describe the transformations that has taken place in each family graph.

More Wide, and right 4 unitsRight 3 units, and up 10 units

More Narrow

Reflected over x-axis, and moved right 5 units

Describe the transformations that has taken place in each family graph.

Reflect over x-axis, and up 2 units

Reflected over y-axis

Right 2 units

FINDING INVERSE FINDING INVERSE FUNCTIONSFUNCTIONS

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of ,

y x 2

x y 2

x y2

y x

f 1(x) x , x 0

f (x) x 2

x 0

FINDING INVERSE FINDING INVERSE FUNCTIONSFUNCTIONS

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of f (x) = 4x + 5

y 4x 5

x 4y 5

x 5 4y

x 54

y

f 1(x) x 5

4

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of f (x) = 2x3 - 1

f 1(x) x 1

23

y 2x 3 1

x 2y 3 1

x 12y 3

x 1

2y 3

y x12

3

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of

Find the inverse of Steps for findingan inverse.

1. solve for x

2. exchange x’sand y’s

3. replace y with f-1

Graph then function and it’s inverse of the same graph.

Parabola shifted 4 units left, and 1 unit down

Now to graph the inverse, just take each point and switch the x and y value and graph the new points.

Ex: (-4, -1) becomes (-1, -4)

Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

Graph then function and it’s inverse of the same graph.

Cubic graph shifted 5 units to the left

Now to graph the inverse, just take each point and switch the x and y value and graph the new points.

Ex: (-5, 0) becomes (0, -5)

Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

Graph then function and it’s inverse of the same graph.

Parabola shifted down 2 units

Now to graph the inverse, just take each point and switch the x and y value and graph the new points.

Ex: (0, -2) becomes (-2, 0)

Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

x 2

f xx 2 x 2

Vertical Asymptotes: x 2Horizontal Asymptotes: y 0

Holes: 1

2,4

Intercepts:

1

x 2

10,

2

10,

2

Determine if each parabola has a maximum value or a minimum value. y = ax2 + bx + c

“a” is positive so that means it opens up, and has a minimum

“a” is negative so that means it opens down, and has a maximum

Graph each inequality:

Find the maximum point of the graph of each:

Find the x and y intercepts of

Without graphing, describe the end behavior of the graph of

Positive coefficient, even power means it rises right and left

Negative coefficient, even power means it falls right and left

positive coefficient, odd power means it rises right and falls left

Without graphing, describe the end behavior of the graph of

Positive coefficient, even power means it rises to left and falls to right

Positive coefficient, odd power means it rises right and falls left

positive coefficient, even power means it rises right and rises left

Part Two

Determine whether each function is even, odd, or neither.

Graph the function Find the inverse equationGraph the inverse on the same graph. Is the inverse a function?

Determine the asymptotes for the rational function then graph it

Graph the inequality

Find the derivative of the function

Find the derivative of the function

Find the equation of the tangent to

y = x3 + 2x at:

A.) x = 2 B.) x = -1

C.) x = -2

f’(x)=0

Step 1: Find the derivative, f’(x)

Step 2: Set derivative equal to zero and solve, f’(x)=0

Step 3: Plug solutions into original formula to find y-value, (solution, y-

value) is the coordinates.

Note: If it asks for the equation then you will write y=y value found when

you plugged in the solutions for f’(x)=0

Determine three critical points that are found on the graph of .Identify each equation as a relative max, min, or point of inflection.

Find the x and y intercept of

Sketch the graph ofDescribe the graph.