Chapter 1 Notes: Functions and Graphs

Name:______

Math Analysis

Day 6: Section 1-1 Graphs

Points and Ordered Pairs

The Rectangular Coordinate System (aka: The Cartesian coordinate system)

Practice: Label each on the coordinate system shown. 1) x-axis, 2) y-axis, 3) origin 4) Quad I, 5) Quad II, 6) Quad III, 7) Quad IV. Also plot the points: A(−2. 4), B(4, −2), C(−3. 0), and D(0, −3).

Graphing Equations by Plotting Points

A relationship between two quantities can be expressed as an equations in two variables, such as:

A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement.

Practice: In 1-2, Determine if the given ordered pairs are solutions to the equation .

1. (10, 96)2. (0, 2)

Graphing an Equation Using the Point-Plotting Method.

Select values for x (for this section you will use the integers from −3 to 3)
Substitute each x-value into the equation and solve for y.
Create order pairs by grouping the x-value with it’sy-value.
Plot the order pairs

Practice: 1. Graphusing the Point-Plotting Method.

x / / Ordered Pair (x, y)
−3
−2
−1
0
1
2
3

Intercepts

x-interceptof a graph is the x-coordinate when the y-coordinate equals zero. We also can describe the x-intercept graphically at the point where the graph intersects the x-axis.
y-intercept of a graph is the y-coordinate when the x-coordinate equals zero. We also can describe the y-intercept graphically at the point where the graph intersect the y-axis.

Practice: 1-5, Identify the x- and y-intercepts of the given graphs or equation.

1. x-int: y-int: 2. x-int: y-int:

3. x-int: y-int: 4. x-int: y-int:

Section 1-2 Basics of Functions and Their Graphs

Relation

Practice: Find the domain and the range of the relation:

Domain:

Range:

A relation of order pairs is a function if all of the domain (x) values are different. A function can have two different domain values with the same range value.

Practice: In 1-2 Determine whether each relation is a function:

1. 2.

Determining Whether an Equation Represents a Function.
To determine whether an equation defines y as a function of x:

Solve the equation in terms of y.
If only one value of y can be obtained for a given x, the equation is a function.

Example: x2 + y = 4

−x2 −x2

y = 4 – x2

If two or more values of y can be obtained for a given x, the equation is not a function.

Example: x2 + y2 = 5

−x2 −x2

y2 = 5– x2

Practice: In 1-2, determine whether each equation defines y as a function of x.

1. 2.

Function Notation

If an equation in x and y gives only one value of y for each value of x, then the variable y is a function of the variable x. We use function notation by replacing ywith . We think of a functions domain (x-components) as the set of the function’s input values and the range (y-components) as the set of the function’s output values. The special notation , read “f of x” represents the value of the function at the number x.

xf(x)

Evaluating a Function

To evaluate a function substitute the input value in for x and evaluate the expression.

Practice: If , evaluate each of the following:

(a) (b) (c)

Graphs of Functions

Practice: Graph the functions and in the same rectangular coordinate system. Select integers for x, starting with −2 and ending with 2. How is the graph of g related to the graph of f?

The Vertical Line Test

Practice: In 1-4, Use the vertical line test to identify graphs in which y is a function of x.

1. 2.

3. 4.

Day 7: Section1-3 More on Functions and There Graphs

The Difference Quotient

To evaluate the difference quotient:

Find the value of by using substitution, replacing x with (x + h) and simplify the expression.
Use the expression found in step one and subtract from it.
Use the expression found in step two and divide by h. This new expression is the called the difference quotient.

Example: If , find and simplify each expression.

(a) (b)

(a) We find by replacing x with x + h

each time that x appears in the equation.

f(x) = 3x2 − 2x + 4

f(x + h) = 3(x + h)2 − 2(x + h) + 4

= 3(x2 + 2xh + h2) – 2x – 2h + 4

= 3x2 + 6xh +3 h2 – 2x – 2h + 4

Practice: In 1-2, Find and simplify the difference quotient: for the given function.

1. 2.

Piecewise Functions

To evaluate a Piecewise Function

Only substitute the x-value into the expression where the inequality of x is true. When you evaluate this expression the result is the value of the function with that given x.

Practice: In 1-2, Evaluate each piecewise function at the given values of the independent variable.

(a) (b) (c)

Practice: In 1-2, State the intervals on which the given function is increasing, decreasing, or constant.

1. 2.

Increasing: Increasing:

Decreasing: Decreasing:

Constant: Constant:

Relative Maxima and Relative Minima

Practice: Use the graph of f to determine each of the following. Where applicable, use interval notation.

1. the domain of f

2. the range of f

3. the x-intercept(s)

4. the y-intercept(s)

5. interval(s) on which f is increasing

6. interval(s) on which f is decreasing

7. intervals(s) on which f is constant

8. the relative minimum of f

9. the relative maximum of f

10. f(−3)

11. the value(s) of x when f(x) = −2

1.Determine the range.

2.Determine the domain.

3.Determine the interval(s) the function is increasing.

4.Determine the interval(s) the function is decreasing.

5.Determine the interval(s) the function is constant.

6.State the relative maximum(s) if any.

7.State the relative minimum(s) if any

8.Evaluate:

Even and Odd Functions and Symmetry

Practice: In 1-3, Determine whether each of the following functions is even, odd, or neither.

1. 2. 3.

Day 8: Section 1-4Linear Functions and Slope; Section 1-5 More on Slope

The Slope of a Line

Practice: In 1-2, Find the slope of the line passing through each pair of points:

1. (−3, 4) and (−4, −2)2. (4, −2) and (−1, 5)

Possibilities for a Line’s Slope

Positive Slope / Negative Slope / Zero Slope / Undefined Slope

Line Rises from left to right /

Line falls from left to right /

Line is horizontal /

Line is vertical

Equations of Lines

1. Point-slope form: / y – y1 = m(x – x1) where m = slope, (x1, y1) is a point on the line
2. Slope-intercept form: / y = mx + b where m = slope, b = y-intercept
3. Horizontal line: / y = b where b = y-intercept
4. Vertical line: / x = a where a = x-intercept
5. General form: / Ax + By + C = 0 where A, B and C are integers and A must be positive.

Practice: In 1-6, use the given conditions to write an equation for each line in (a) point-slope form, (b) slope-intercept form and (c) General form.

1. Slope = 8, passing through (4, −1)2. Slope = , passing through

3. Passing through (−3, −2) and (3, 6)4. Passing through

5. x-intercept = 3 and y-intercept = 16. x-intercept = and y-intercept =

Parallel and Perpendicular Lines

Practice: Write an equation of the line passing through (−2, 5) and parallel to the line whose equation is y = 3x + 1. Express the equation in point-slope from and slope-intercept form.

Practice:

Day 9: Section 1-6Transformations of functions

Algebra’s Common Graphs

Summary of Transformations (In each case, c represents a positive real number.)

To Graph / Draw the Graph of f and: / Changes in the Equation of y=f(x)
Vertical Shifts
y = f(x) + c
y = f(x) – c / Raise the graph of f by c units
Lower the graph of f by c units / c is added to f(x)
c is subtracted from f(x)
Horizontal Shifts
y = f(x + c)
y = f(x – c) / Shift the graph of f to the left c units
Shift the graph of f to the right c units / x is replaced with x + c
x is replaced with x − c
Reflection about the x-axis
y = −f(x) / Reflect the graph of f about the x-axis / f(x) is multiplied by −1
Reflection about the y-axis
y = f(−x) / Reflect the graph of f about the y-axis / x is replaced with −x
Vertical Stretching or Shrinking
y = cf(x), c > 1
y = cf(x), 0 < c < 1 / Multiply each y-coordinate of y = f(x) by c, vertically stretching the graph of f.
Multiply each y-coordinate of y = f(x) by c, vertically shrinking the graph of f. / f(x) is multiplied by c, c > 1
f(x) is multiplied by c, 0 < c < 1
Horizontal Stretching or Shrinking
y = f(cx), c > 1
y = f(cx), 0 < c < 1 / Divide each x-coordinate of y = f(x) by c, horizontally shrinking the graph of f.
Divide each x-coordinate of y = f(x) by c, horizontally stretching the graph of f. / x is replaced with cx, c > 1
x is replaced with cx, 0 < c < 1

Practice: Graph the standard function f(x) and then graph the given function. Describe the transformations need to change the common function f(x) to get g(x).

1. g(x) = −(x + 3)2 + 42. g(x) =

Section 1-7Combinations of Functions and Composite Functions;

Finding the Domain of a Function

Practice: In 1-3, Use interval notation to express the domain of each function:

1. 2. 3.

Practice: Let and. Find each of the following functions and determine the domain:

1. (f + g)(x)2. (f – g)(x)3. (fg)(x)4.

Composite Functions

Practice: Given f(x) = 5x + 6 and g(x) = 2x2 – x – 1, find each of the following composite functions:

1. 2. 3.

Geometry Review: Trigonometry

A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent. These three rations are defined for the acute angles of right triangles, though your calculator will give you values of sine, cosine, and tangent for angles of greater measure. The abbreviations for the ratios are sin, cos, and tan respectively.

In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in reduced fractional form.

1. sinM 2. cosZ
3. tanL 4. sinX
5. cosL 6. tanZ /

In 7-10, find the trigonometric ratio that corresponds to each value and the angle given, using the triangle at the right.

7. 8.
9. 10. /

Day 11:Section 1-8 Inverse Functions

Inverse Functions

Finding the Inverse of a Function

The equation for the inverse of a function f can be found as follows:

Replace f(x) with y in the equation for f(x).
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and his procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 with . We can verify our result by showing that and .

Practice: Find the inverse of each function.

1. 2. 3.

The Horizontal Line Test and One-to-One Functions

Practice: In 1-4 Which of the following graph represent functions that have inverse functions?

1. 2. 3. 4.

Graph of f and f−1

To graph an inverse function given the graph of ordered pairs of the function

Find an ordered pair on the function.
To graph the inverse just take the x-coordinate of f(x) is the y-coordinate of f−1(x) and the y-coordinate of f(x) is the x-coordinate of f−1(x).
Continue finding order pairs on f(x) and interchange the x- and y-coordinates to plot points on f−1(x).
Connect points with a smooth curve.

Practice: Use the graph of f to draw the graph of its inverse function.

Section 1-9 Distance and Midpoint Formulas; Circles

The Distance Formula

Practice: Find the distance between the two points given.

1. (−4, 9) and (1, −3)2.

The Midpoint Formula

Practice: Find the midpoint of the line segment with endpoints at: .

Geometry Review: Trigonometry

Besides the three most common trigonometric ratios, sine, cosine, and tangent, there are three more rations that are considered the reciprocal ratios. These reciprocal ratios are cosecant, secant, and cotangent. The abbreviations for the ratios are csc, sec, and cot respectively.

In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in reduced fractional form.

1. cscM 2. secZ
3. cotL 4. cscX
5. secL 6. tanZ /

How to find another trigonometric equation given one trigonometric equation.

Use the given information to draw a right triangle and making the given sides
Find the missing side using Pythagorean Theorem
Now that you have all there sides of the right triangle labeled you can write the trigonometric equation for any ratio.

In 7-: Use the given trig equation to find the value of a different trig ratio.

7.) 8.)

9.) 10.)

Day 12: Section 1-10 Modeling with Functions

Modeling with Functions; Word Problems

Practice:

1. A car rental agency charges $200 per week plus $0.15 per mile to rent a car.

(a) Express the weekly cost to rent the car, f, as a function of the number of miles driven during the week, x.

(b) How many miles did you drive during the week if the weekly cost to ret the car was $320?

2. The bus fare in a city is $1.25. People who use the bus have the option of purchasing a monthly coupon book for $21.00. With the coupon book, the fare is reduced to $0.50.

(a) Express the total monthly cost to use the bus without a coupon book, f, as a function of the number of times in a month

the bus is used, x.

(b) Express the total monthly cost to use the bus with a coupon book, g, as afunction of the number of times in a month

the bus is used, x.

is the same as the total monthly cost with the coupon book. What will the monthly cost for each option?

Chapter 1 Review Sheet

Complete each problem on a separate sheet of paper. All graphs must be on graph paper!!Show all of your work! NO WORK = NO CREDIT!!

1. Find the inverse of; a. b.

For questions 2-3, given , find the following:

2. f(x + 2)3. f(8)

For questions 4-6, find the domainof each function. Write your answer in interval notation.

4. 5. 6.

For questions 7-10 , using the correct terminology, describe the transformations, in words, from the graphs of the parent functions, f(x), to the graphs of g(x). Then, graph g(x). Make sure to label your axes completely! Please use graph paper for the graphs

7. 8.

9. 10.

For 11-13, let , , and . Find the indicated values.

11. 12. 13.

In 14 -15, find and simplify the difference quotient for the given function.

14. 15.

In 17-18, write the equation of the line in (a) point slope form (b) slope intercept form and (c) general form given the following information.

17. passing through (1, 5)18. passing through

parallel to 2x + 5y – 4 =0

In 19-21, determine if the function is odd, even, or neither. If graph of the function has symmetry state so.

19. 20. 21.

For questions 22-23, use the graph below. Write answers, were appropriate, in interval notation.

22. Determine the range.

23. Determine the domain.

24. Determine the interval(s) the function is increasing.

25. Determine the interval(s) the function is decreasing.

26. Determine the interval(s) the function is constant.

27.State the relative maximum(s) if any.

28.State the relative minimum(s) if any.

29. State the x-intercept(s)

30. Sate the y-intercept(s)

31. Find

For #21-26, find the value of each variable. If your answer is not an integer, leave it in simplest radical (root) form.

21.
/ 22.
/ 23.

24.
/ 25.
/ 26.

For #27-29, determine if the lengths of the sides of a triangle given from a right triangle.

27. / 28. 10, 15, 12 / 29. 41, 9, 40

For questions 30-31, draw a diagram and solve for the missing information. Leave your answer in simplest radical (root) form.

30. The perimeter of an equilateral triangle is 33 cm. Find the length of the height of the triangle.

31. A square has a 60-cm diagonal. How long is each side of the square?

Trig Worksheet #2Name:

Period:

In 1-6, Use the given trigonometric equation to find the remaining five trigonometric equations.

1. / 2. / 3.
4. / 5. / 6.

In7-12, Use the given trigonometric equation to find the indicated trigonometric ratio.

7. / 8. / 9.
10. / 11. / 12.

Trig Worksheet #1Name

Period

For 1-2,Write the sine, cosine, & tangent ratio for

1.)
/ 2.)

3.) In the triangle below, if what is ?
/ 4.) In the triangle below, if what is and ?

In5-10, find the indicated trigonometric ration using the right triangles to the right. Final answers should be in reduced fractional form.

5. sin J 6. Cos K
7. tan C 8. Sin S
9. cos J 10. Tan K /

Name:______Right Triangle Worksheet #1

Find the value of each variable. Leave your answers in simplest radical form.

3. 4.

Determine if the numbers represent the lengths of the sides of a right triangle. Show work to justify why or why not.

5. 6, 9, 10

6. 7, 24, 25

7. 18, 24, 30

Geo Worksheet #5Name

Special Right Triangles

For #1-6, find the value of each variable. If your answer is not an integer, leave it in simplest radical (root) form.

1.
/ 2.
/ 3.

4.
/ 5.
/ 6.

7. / 8. / 9.
10.) A right triangle’s hypotenuse has length 5. If one leg has length 2, what is the length of the other leg?
A. 2
B. 3
C.
D.
E. 7 / 11.) What is the value of x in the triangle below?

A. 5
B.
C.
D. 10
E. 20

Geo. Worksheet #4Name

Special Right Triangle Review

Find the value of each variable. If your answer is not an integer, leave it in simplest radial form.

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

7.) Find the length of a diagonal of a square with sides of 10 inches long.

8.) One side of an equilateral triangle measures 12cm. Find the measure of an altitude of the triangle.