5-1 Linear Equations and Functions Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

5-1 Linear Equations and Functions

Warm UpWarm Up

Lesson Presentation

California Standards

PreviewPreview


2. Evaluate the function f(x) = for –10, –5, 0, 5, and 10.

f(–10) =

f(–5) =

f(0) =

f(5) =

f(10) =

Warm Up

1. Solve 2x – 3y = 12 for y.

–1

0

1

2

3


California Standards

6.0 Students graph a linear equation and compute x- and y- intercepts (e.g. graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequalities (e.g., they sketch the region defined by 2x + 6y < 4). Also covered: 7.0, 17.0, 18.0


linear equationlinear function

Vocabulary


Many stretches on the German autobahn have a speed limit of 120 km/h. If a car travels continuously at this speed, y = 120x gives the number of kilometers y that the car would travel in x hours.

Notice that the graph is a straight line. An equation whose graph forms a straight line is a linear equation. Also notice that this is a function. A function represented by a linear equation is a linear function.


For any two points, there is exactly one line that contains them both. This means you need only two ordered pairs to graph a line. However, graphing three points is a good way to check that your line is correct.

5-1 Linear Equations and FunctionsAdditional Example 1A: Graphing Linear EquationsGraph y = 2x + 1. Tell whether it represents a function.

Step 1 Choose three values of x and generate ordered pairs.

1

0

–1

y = 2(1) + 1 = 3 (1, 3)

y = 2(0) + 1 = 1

y = 2(–1) + 1 = –1

(0, 1)

(–1, –1)

Step 2 Plot the points and connect them with a straight line. No vertical line will intersect this graph more than once. So y = 2x + 1 describes a function.

x y = 2x + 1 (x, y)


Sometimes solving for y first makes it easier to generate ordered pairs using values of x. To review solving for a variable, see Lesson 2-6.

Helpful Hint


Additional Example 1B: Graphing Linear Equations

Graph 15x + 3y = 9. Tell whether it represents a function.

Step 1 Solve for y.

15x + 3y = 9–15x –15x

3y = –15x + 9

y = –5x + 3

Subtract 15x from both sides.

Since y is multiplied by 3 divide both sides by 3.


Step 2 Choose three values of x and generate ordered pairs

Additional Example 1B Continued

Step 3 Plot the points and connect them with a straight line. No vertical line will intersect this graph more than once. So 15x + 3y = 9 describes a function.

Graph 15x + 3y = 9. Tell whether it represents a function.

x y = –5x + 3 (x, y)

1

0

–1

(1, –2)

(0, 3)

(–1, 8)

y = –5(1) + 3 = –2

y = –5(0) + 3 = 3

y = –5(–1) + 3 = 8

5-1 Linear Equations and FunctionsAdditional Example 1C: Graphing Linear Equations

Graph x = –2. Tell whether it represents a function.

Any ordered pair with an x-coordinate of –2 will satisfy this equation.

There is a vertical line that intersects this graph more than once, so x = –2 does not represent a function.

Plot several points that have an x-coordinate of –2 and connect them with a straight line.

5-1 Linear Equations and FunctionsAdditional Example 1D: Graphing Linear Equations

Graph y = 8. Tell whether it represents a function.

Any ordered pair with a y-coordinate of 8 will satisfy this equation.

Plot several points that have an y-coordinate of 8 and connect them with a straight line.

No vertical line will intersect this graph more than once, so y = 8 represents a function.

5-1 Linear Equations and FunctionsCheck It Out! Example 1a

Graph y = 4x. Tell whether it represents a function.


1

0

–1

(1, 4)

(0, 0)

(–1, –4)

y = 4(1) = 4

y = 4(0) = 0

y = 4(–1) = –4

Step 2 Plot the points and connect them with a straight line. No vertical line will intersect this graph more than once. So y = 4x describes a function.

x y = 4x (x, y)


Check It Out! Example 1b

Graph y + x = 7. Tell whether it represents a function.

Step 1 Solve for y.

y + x = 7 –x –x

y = –x + 7

Subtract x from both sides.

5-1 Linear Equations and FunctionsCheck It Out! Example 1b Continued

Graph y + x = 7. Tell whether it represents a function.


Step 3 Plot the points and connect them with a straight line. No vertical line will intersect this graph more than once. So y + x = 7 describes a function.

x (x, y)

1

0

–1

(1, 6)

(0, 7)

(–1, 8)

y = –(1) + 7 = 6

y = –x + 7

y = –(0) + 7 = 7

y = –(–1) + 7 = 8

5-1 Linear Equations and FunctionsCheck It Out! Example 1c

Graph . Tell whether it represents a function.

There is a vertical line that intersects this graph more than once, so x = does not describe a function.

Plot several points that have an x-coordinate of and connect them with a straight line.

Any ordered pair with an x-coordinate of will satisfy this equation.



Additional Example 2A: Determining Whether a Point is on a Graph

Without graphing, tell whether each point is on the graph of 2x + 5y = 16.

(3, 2)

Substitute: 2x + 5y = 16

16 = 16

Since (3, 2) is a solution to 2x + 5y = 16, (3, 2) is on the graph.

2(3) + 5(2) 16=?

6 + 10 16=?


Additional Example 2B: Determining Whether a Point is on a Graph

Without graphing tell whether each point is on the graph of 2x + 5y = 16.

(2, 2)


14 16

Since (2, 2) is not a solution to 2x + 5y = 16, (2, 2) is not on the graph.

2(2) + 5(2) 16=?

4 + 10 16=?


Additional Example 2C: Determining Whether a Point is on a Graph

Without graphing tell whether each point is on the graph of 2x + 5y = 16.

(8, 0)


16 = 16

Since (8, 0) is a solution to 2x + 5y = 16, (8, 0) is on the graph.

2(8) + 5(0) 16=?

16 + 0 16=?


Check It Out! Example 2a

Without graphing tell whether each point is on the graph of x – 3y = 12.

(5, 1)

Substitute: x – 3y = 12

Since (5, 1) is not a solution to x – 3y = 12, (5, 1) is not on the graph.

2 12

5 – 3(1) 12=?

5 – 3 12=?




(0, –4)


12 = 12

Since (0, –4) is a solution to x – 3y = 12, (0, –4) is on the graph.

0 – 3(–4) 12=?

0 + 12 12=?


Check It Out! Example 2c


(1.5, –3.5)


12 = 12

Since (1.5, –3.5) is a solution to x – 3y = 12, (1.5, –3.5) is on the graph.

1.5 – 3(–3.5) 12=?

1.5 + 10.5 12=?


Linear equations can be written in the standard form as shown below.


Notice that when a linear equation is written in standard form.

• x and y both have exponents of 1.

• x and y are not multiplied together.• x and y do not appear in denominators,

exponents, or radical signs.



Additional Example 3A: Writing Linear Equations in Standard Form

Write x = 2y + 4 in standard form and give the values of A, B, and C. Then describe the graph.

x = 2y + 4–2y –2y

x – 2y = 4

Subtract 2y from both sides.

The equation is in standard form.

A = 1, B = –2, C = 4

The graph is a line that is neither horizontal nor vertical.


Additional Example 3B: Writing Linear Equations in Standard Form

Write x = 4 in standard form and give the values of A, B, and C. Then describe the graph.

x + 0y = 4

A = 1, B = 0, C = 4

The equation is in standard form.

The graph is a vertical line at x = 4.

x = 4


Check It Out! Example 3a

Subtract 5x from both sides.The equation is in standard form.

y = 5x – 9–5x –5x

–5x + y = – 9

Write y = 5x – 9 in standard form and give the values of A, B, and C. Then describe the graph.

A = –5, B = 1, C = –9

The graph is a line that is neither horizontal nor vertical.



y = 12

Write y = 12 in standard form and give the values of A, B, and C. Then describe the graph.

0x + y = 12 The equation is in standard form.

A = 0, B = 1, C = 12

The graph is a horizontal line at y = 12.


Check It Out! Example 3c

Write x = 2 in standard form and give the values of A, B, and C. Then describe the graph.

x = 2

x + 0y = 2 The equation is in standard form.

A = 1, B = 0, C = 2

The graph is a vertical line at x = 2.


• y – x = y + (–x)

• y +(–x) = –x + y

• –x = –1x

• y = 1y

Remember!


For linear functions whose graphs are not horizontal, the domain and range are all real numbers. However, in many real-world situations, the domain and range must be restricted. For example, some quantities cannot be negative, such as distance.


Sometimes domain and range are restricted even further to a set of points. For example, a quantity such as number of people can only be whole numbers. When this happens, the graph is not actually connected because every point on the line is not a solution. However, you may see these graphs shown connected to indicate that the linear pattern, or trend, continues.

5-1 Linear Equations and FunctionsAdditional Example 4: Application

The relationship between human years and dog years is given by the function y = 7x, where x is the number of human years. Graph this function and give its domain and range.

x (x, y)

2 (2, 14)

(6, 42)

y = 7(2) = 14

y = 7x

y = 7(4) = 28

y = 7(6) = 42

4

6

(4, 28)

Choose several values of x and make a table of ordered pairs.

The ages are continuous starting with 0, so the domain is: {x ≥ 0} and the range is: {y ≥ 0}.


Graph the ordered pairs.

Additional Example 4 Continued

(2, 14)

(4, 28)

(6, 42) Any point on the line is a solution in this situation. The arrow shows that the trend continues.

Human Years vs. Dog Years


Check It Out! Example 4

What if…? At another 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) = 3x + 10, where x is the number of manicures. Graph this function and give its domain and range.


Check It Out! Example 4 Continued

The number of manicures must be a whole number, so the domain is {0, 1, 2, 3, …}. The range is {10, 13, 16, 19, …}.

Choose several values of x and make a table of ordered pairs.

x f(x) = 3x + 10

0 f(0) = 3(0) + 10 = 10

1 f(1) = 3(1) + 10 = 13

2 f(2) = 3(2) + 10 = 16

3 f(3) = 3(3) + 10 = 19

4 f(4) = 3(4) + 10 = 22

5 f(5) = 3(5) + 10 = 25


Check It Out! Example 4 Continued

Graph the ordered pairs.

The individual points are solutions in this situation. The line shows that the trend continues.


Lesson Quiz: Part I

Graph each linear equation. Then tell whether it represents a function.

1. 2y + x = 6 2. 3y = 12

Yes, it is a function. Yes, it is a function.


Lesson Quiz: Part II

Without graphing, tell whether each point is on the graph of 6x – 2y = 8.

3. (1, 1) 4. (3, 5)no yes

5. The cost of a can of iced-tea mix at SaveMore Grocery is $4.75. The function f(x) = 4.75x gives the cost of x cans of iced-tea mix. Graph this function and give its domain and range.

D: {0, 1, 2, 3, …}

R: {0, 4.75, 9.50, 14.25, …}

5-1 Linear Equations and Functions Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

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