Algebra 1 Topic 4 Relations Functions Table of

Algebra 1 Topic 4: Relations & Functions

Table of Contents 1. 2. 3. 4. 5. 6. Using Graphs to Relate Quantities Patterns and Linear Functions Nonlinear Functions Graphing a Function Rule Key Features of Graphs Formalizing Relations & Functions

Using Graphs to Relate Quantities

Miami Traffic Which graph could show a car sitting at a stoplight? How do you know?

Connecting Graphs to Stories Each day several leaves fall from a tree. One day a gust of wind blows off many leaves. Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation.

Let’s Practice… The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation.

Let’s Practice… Marcus and Janine made the table shown below to represent the difference between their ages during different years. Which graph matches the information in the table?

Let’s Practice… The table shows the total number of customers at a car wash after 1, 2, 3, and 4 days of its grand opening. Which graph could represent the data shown in the table?

Let’s Practice… This graph shows someone taking a walk in the neighborhood. Describe what it shows by labeling each part.

Draw Your Own Graph… A pelican flies above the water searching for fish. Sketch a graph of its altitude from takeoff from shore to diving to the water to catch a fish.

Patterns and Linear Functions

Vocab Independent Variable – A variable that doesn’t depend on any other value. – Generally, x is used for the independent variable. Dependent Variable – A variable whose value depends on some other value. – Generally, y is used for the dependent variable.

Identify the Independent and Dependent Variables… 1. A painter must measure a room before deciding how much paint to buy. 2. The height of a candle decrease d centimeters for every hour it burns. 3. A veterinarian must weight an animal before determining the amount of medication.

Vocab Input – Values of the independent variable. Output – Values of the dependent variable. Independent Variable Dependent Variable

Vocab Function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value. Function Rule Output variable Input variable

Representing Linear Relationships Step 1: Make a table. Step 2: Look for a pattern in the table. Step 3: Create an equation to represent the pattern. Step 4: Use the table to make a graph.

What’s the Function? A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. The table below represents this situation. Number of DVDs purchased x 0 1 2 3 4 5 Total cost ($) y 20 35 50 65 80 95

Let’s Practice… The table shows the amount of water y in a tank after x minutes of being drained. 1. Is the relationship function? 2. Describe the relationship using words. 3. Write an equation for the relationship.

Nonlinear Functions

Nonlinear Function: a function whose graph is not a line or part of a line.

Let’s Practice… Is the graph is linear or nonlinear? A. B.

Let’s Practice… Is the graph is linear or nonlinear? C. D.

What’s Special About Nonlinear Functions? !? ! • Linear functions have a constant relationship between x and y. • HOWEVER, nonlinear functions, the differences between x-values and the corresponding yvalues are not the same. • Yet, there still is a pattern….

Let’s Practice… What’s the pattern? ? Tell whether the function in the table has a linear or nonlinear relationship.

Let’s Practice… What’s the pattern? ? Tell whether the function in the table has a linear or nonlinear relationship. Input Output 1 2 2 5 3 11

Writing a Rule to Describe a Nonlinear Function Step 1: Make a table to organize the x and y values. Step 2: Look for a pattern in the y-values and identify a rule that produces the given y-value when you substitute the x-value. Step 3: Write and verify the function rule.

Let’s Practice… Write a function rule for the nonlinear function: x y 1 2 2 4 3 8 4 16 5 32

Let’s Practice… The ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25) represent a function. What is a rule that represents this function?

Graphing a Function Rule

Vocab Discrete: A discrete function is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, . . . Continuous: A continuous random variable is one which takes an infinite number of possible values. • Continuous random variables are usually measurements. Car Approaching Traffic Light

Discrete or Continuous? Henry begins to drain a water tank by opening a valve. Water Level Water tank Time Jamie is taking an 8 -week keyboarding class. At the end of each week, she takes a test to find the number of words she can type per minute.

Graphing a Line 1. Pick three x-values. • One should be negative, one should be Procedure: Graphing Functions zero and one should be positive. 2. Plug in the x-values to the function to determine y value. 3. Graph each ordered pair. 4. Connect the dots with a line.

Let’s Practice… Graph y = 2 x + 1. X Y

Let’s Practice… Graph y = 4 x. X Y

Let’s Practice… Graph 15 x + 3 y = 9. X Y

Dog Years… The relationship between human years and dog years is given by the function y = 7 x, where x is the number of human years. Graph this function. X Y

Let’s Practice… At a salon, Sue can rent a station for $10. 00 per day plus $3. 00 per manicure. The amount she would pay each day is given by f(x) = 3 x + 10, where x is the number of manicures. Graph this function. X Y

Graphing Nonlinear Functions 1. Pick 5 x-values. • Negatives, positives and zero. 2. Plug in the x-values to the function to determine y-value. 3. Graph each ordered pair. 4. Connect the dots with a line.

Let’s Practice… Graph the function f(x) = |x| + 2. X Y

Let’s Practice… Graph the function y = |x – 1|. X Y

Let’s Practice… Graph the function y = x 2. X Y

Let’s Practice… Graph the function y = x 2 + 1. X Y

Key Features of Graphs

Key Features of Graph Intercepts Relative Extremes Symmetry Slope Form X- Intercepts Y- Intercepts Relative Maximum Relative Minimum y-axis x-axis y=x Positive Negative Increasing Decreasing

Intercepts X-intercept: Where the graph crosses the x axis. The y-value is zero. Y-intercept: Where the graph cross the y axis. The x-value is zero.

Let’s Practice… What are the x-intercept and y-intercept?

Relative Extremes Relative Maximum: highest point on the graph around it (looks like a hill) Relative Minimum: lowest point on the graph around it (looks like a valley)

Let’s Practice… What is a relative maximum of this graph? What is a relative minimum of this graph?

Symmetry x-axis: When flipped over the x-axis, the graph looks the same E

Symmetry y-axis: When flipped over the y-axis, the graph looks the same

Symmetry y=x: When flipped over the line y=x, the graph looks the same

Let’s Practice… What kind of symmetry does the picture have?

Slope Positive: the graph increases from left to right Negative: the graph decreases from left to right Negative Positive

Let’s Practice… Of lines A, B, and C, which lines are positive? Which lines are negative? B A C

Form Increasing: The values of y get bigger from left to right Decreasing: The values of y get smaller from left to right

Let’s Practice… Name the internals where the function is increasing and decreasing.

Formalizing Relations and Functions

Relation – A relationships that can be represented by a set of ordered pairs. Table x y 2 3 4 7 6 8 Example: Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. y x 2 3 4 7 6 8

Domain & Range Domain: The set of first coordinates (or xvalues) of the ordered pairs of a relation. Range: The set of second coordinates (or y-values) of the ordered pairs of a relation.

Example: Finding Domain and Range from a Graph y = x 2 Only nonnegative yvalues appear on the graph. All x-values appear somewhere on the graph. For y = x 2 the domain is all real numbers and the range is y ≥ 0.

Let’s Practice… Give the domain and range of the relation.

Let’s Practice… Give the domain and range of the relation. x 1 4 8 y 6 1 5 4 2 1 1 – 4 – 1 0

Functions Function: a special type of relation that pairs each domain value with exactly one range value. All functions are relations, but all relations are not functions.

Is This Relation a Function? Give the domain and range of the relation. Tell whether the relation is a function. Explain. 1. {(3, – 2), (5, – 1), (4, 0), (3, 1)} 2. {(-1, 5), (0, 2), (1, 3), (5, 7)} 3. {(3, 9), (4, -6), (0, 9), (-1, -5)}

Is This Relation a Function? Give the domain and range of the relation. Tell whether the relation is a function. Explain. 1. {(3, – 2), (5, – 1), (4, 0), (3, 1)} D: {3, 4, 5} R: {-2, -1, 0, 1} No. The 3 connects to both the -2 and 1. 2. {(-1, 5), (0, 2), (1, 3), (5, 7)} D: {-1, 0, 1, 5} R: {2, 3, 5, 7} Yes, since each domain value does not repeat. 3. {(3, 9), (4, -6), (0, 9), (-1, -5)} D: {-1, 0, 3, 4} R: {-5, -6, 9} Yes, since each domain value does not repeat.

Is This Relation a Function? Give the domain and range of the relation. Tell whether the relation is a function. Explain. – 4 – 8 4 5 2 1

Let’s Practice… Give the domain and range of the relation. Tell whether the relation is a function. Explain. a. {(8, 2), (– 4, 1), (– 6, 2), (1, 9)} b.

Let’s Practice… Give the domain and range of each relation. Tell whether the relation is a function and explain.

Short Cut: Vertical Line Test

Let’s Practice… Graph each equation. Then tell whether the equation represents a function. y = 3 x – 2 X Y

Let’s Practice… Graph each equation. Then tell whether the equation represents a function. y = |x – 1| X Y

Function Notation f(x) is a fancy way of saying “y”

Evaluating Functions & Finding the Range • We can evaluate a function and find the range by plugging in! Example: Find the range: f(x) = 3 x + 2 using (1, 2, 3, 4)

Let’s Practice… Find the range: 1. f(x) = 2 x -7 (-2, -1, 0, 1 , 2) 2. h(x) = x 2; (-5, 0, 3, 4, 7)