Name:______

Unit 3 – Functions & Graphing

What is a relation?

A ______is a set of ordered pairs.

When you graph a relation there is an independent variable and a ______variable. The ______stands alone. The ______depends on the value of the independent variable.

EX 1: In general, the average price of gasoline slowly and steadily increases through out the year. Identify the independent variable and the dependent. Why?

EX 2: The profit that a business makes generally increases as the price of their product increases. Identify the independent variable and the dependent. Why?

EX 3: In warm climates, the average amount of electricity used in homes each month rises as the daily average temperature increases, and falls as the daily average temperature decreases. Identify the independent and dependent variables. Why?

What is a function?

A ______is a relation in which each element of the ______(x-values) is paired with exactly one element of the ______( y-values)

Or

In a ______, the output depends on the input. There is exactly ______output for each input.

Determine whether each relation is a function. Explain.

EX 1:EX 2:

EX 3: EX 4:

x / y
-3 / 6
2 / 5
3 / 1
2 / 4

EX 5:EX 6:

x / y
-7 / -12
-4 / -9
2 / -3
5 / 0

EX 7:EX 8:

{(-2,4), (1,5), (3,6), (5,8), (7, 10)}{(-5,2), (-2, 5), (0,7), (0,9)}

Vertical Line Test

You can use the ______to see if a graph represents a function. If no vertical line can be drawn so that it ______the graph more than once, then the graph is a function. If a vertical line can be drawn so that it intersects the graph at ______or more points, the relation is not a function.

Determine whether the following are functions. Justify

EX 1: EX 2:

EX 3:EX 4:

Graphing Lines Using a Table of Values

  • A is an equation of a line.
  • Any pair of numbers (x,y) that will work in the equation lie on the line.
  • The pair is a ______to the equation.
  • In equations there are two variables:

and

  • Independent variable:(x) (cause) the values that are controlled or selected
  • Dependent variable:(y) (effect) the values that are a result of the independent variable

EX 1: Think of 2 numbers that add up to 6. If x and y represent these 2 numbers, then the equation

x + y = 6 shows the relationship. Notice, there are infinite number of solutions. Some of these

x / y

solutions are:

This line represents all of the of the equation x + y = 6.

EX 2: Guilderland teachers discovered that if you take the number of example problems tried before a

test, multiply that number by 5 and subtract three this will tell them what their students grade on the test will be.

  1. Write an equation that fits this model.
  2. Graph the equation using a table.
  3. Determine the independent and dependent variables.

EX 3: Courtney has a cell phone plan that charges $25 a month for service plus 10 cents per phone call.

1.Write an equation that fits this model.

2.Graph the equation using a table.

3. Determine the independent and dependent variables.

Graphing

  1. Solve for y in terms of x.
  2. Find at least 3 solutions of the equation by choosing values for x (independent variable) and computing corresponding values for y (dependent variable). Write these answers in a table.
  3. Graph the ordered pairs found in step 2.
  4. Draw a line through the points. Label your line with its equation.

EX 1: Graph the equation of y – x = 3. EX2: Graph the equation 2x + y = 4.

EX 3: Graph the equation EX 4:

Graphing using your Calculator

1. Solve for y in terms of x.

2. Hit the y = key on the calculator.

3. Type the equation in the calculator.

4. Hit 2nd graph. Write down the table of values.

5. Graph and label.

6. Check your graph by using zoom standard

EX1: Using your calculator, graph the EX2: Using your calculator, graph

equation 2.5x + y = 20.the equation 2x – 3y = 6.

Slope

  • The of a line is a number determined by any two points on the line.
  • This number describes how the line is.
  • Slope is the ratio of the change in to the change in .

EX1: Find the slope of the line passing through the points (2,3) and (5,8).

EX2: Find the slope of the line passing through the points (-2,3) and (4,-3).

EX3: Find the slope of the line passing through the points (-4,-2) and (3,8).

EX4: Find the slope of the line passing through the points (3,2) and (3,4). What do you notice?

EX5: Find the slope of the line passing through the points (0,5) and (4,5). What do you notice?

EX6: Find the slope of the given line. EX 7: Find the slope of the given line.

Y y

` xx

EX8: Find the slope of the given line. EX9: Find the slope of the given line.

Y y

xx

Real World Example: At the end of 2 years of teaching, I only had 12 gray hairs. Now, after 10 years

of teaching I have 60 gray hairs.

  1. Identify the independent and dependent variables.
  1. Explain the “real world” significance of the slope.

Graphing a line when given the slope and a point on the line:

EX 1: Draw a line with a slope of that EX 2: Draw a line with a slope of -½

passes through the point (-4,5). that passes through the point (4, -2).

EX 3: Draw a line with a slope of -2 that passes through the point (1, -3).

Graphing Lines Parallel to an Axis

EX 1: Graph the line y = 4

So, the line y= any number is ______

to the ______.

EX 2: Graph the line x = 3

So, the line x = any number is ______

to the ______.

The ______is the place where the graph crosses the y-axis. This means that the

x-coordinate value is equal to ______.

The ______is the place where the graph crosses the x-axis. This means that the

y-coordinate value is equal to ______.

EX 3: Write the equation of the line that is parallel to the y-axis and whose x-intercept is 6.

EX4 : Write the equation of the line that is parallel to the x-axis and whose y-intercept is -3.

Parallel and Perpendicular Lines

Find the slope of the two lines shown on the graph

**Two non-vertical lines are parallel if and only if their slopes are equal.**

Find the slope of the two lines shown on the graph.

**Two non-vertical lines are perpendicular if and only if the slope of one line is the negative reciprocal of the slope of the other line.**

Slope Intercept Form

Given 2x – y = 1

We solve for y: y = 2x – 1

  1. The coefficient of x is the slope (in this case, it’s 2).
  2. The constant term is the y-intercept (in this case, it’s -1).

In general, the equation of a line is given by:

Where m represents the slope and b is the y-intercept.

EX 1:Write the equation of the line whose slope is ½ and whose y-intercept is -4.

EX 2: What is the y-intercept of y = 3x – 5?

EX 3: What is the slope of y = -½ x + 6?

EX 4: What are the slope and y-intercept of the line 2y + x = 14?

EX5: Find the slope and y-intercept of the line 6x – 2y= 10?

EX 6: Draw the graph of EX 7: Draw the graph of 2x+3y= 9.

EX 8: Draw the graph of EX 9: Draw the graph of 3x + 4y = 12.

EX 10: Draw the line through (3,5) whose slope is . Write the equation of the line shown.

Writing the Equation of a Line

Remember: The equation of a line is y = mx + b.

We need 2 things to write the equation of a line

  1. Slope
  2. Y – intercept

EX 1: Write an equation of the line that has a slope of 4 and that passes through the point (3, 5).

EX 2: Write an equation of the line that passes through (2,5) and (4,11).

EX 3: Write an equation of the line that has a slope of ½ and that passes through the point (4,2).

EX 4: Write an equation of the line that passes through (3,1) and (9, -7).

EX 5: Write the equation of the line that is parallel to y = 2x – 4 with a y–intercept of 7.

EX 6: Write the equation of the line that is perpendicular to 2y – 4x = 8 and passes through the point (6,-6).

Graphing Inequalities-Graphing inequalities is very similar to graphing lines. We have to remember:

  1. Whether it is a dashed or solid line.
  2. Whether to shade above or below the line.

It will be a solid line when we have the symbol ____ or _____. It will be a dashed line when we have the symbol _____ or _____. You shade above when _____ or ______and shade below when _____ or ______. Remember this will only work if the variable is on the ______side.

EX1: Graph y  2x.EX2: Graph y – 2x  2.

EX 3: Graph y  3. EX4: Graph 6 - y  3x

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