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5-1 Linear Equations and Functions
Warm UpWarm Up
Lesson Presentation
California Standards
PreviewPreview
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
linear equationlinear function
Vocabulary
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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)
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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.
Step 1 Choose three values of x and generate ordered pairs
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)
5-1 Linear Equations and Functions
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 2 Choose three values of x and generate ordered pairs
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.
5-1 Linear Equations and Functions
5-1 Linear Equations and Functions
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=?
5-1 Linear Equations and Functions
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)
Substitute: 2x + 5y = 16
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=?
5-1 Linear Equations and Functions
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)
Substitute: 2x + 5y = 16
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=?
5-1 Linear Equations and Functions
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=?
5-1 Linear Equations and Functions
Check It Out! Example 2b
Without graphing tell whether each point is on the graph of x – 3y = 12.
(0, –4)
Substitute: x – 3y = 12
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=?
5-1 Linear Equations and Functions
Check It Out! Example 2c
Without graphing tell whether each point is on the graph of x – 3y = 12.
(1.5, –3.5)
Substitute: x – 3y = 12
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=?
5-1 Linear Equations and Functions
Linear equations can be written in the standard form as shown below.
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
Check It Out! Example 3b
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.
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
• y – x = y + (–x)
• y +(–x) = –x + y
• –x = –1x
• y = 1y
Remember!
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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}.
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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.
5-1 Linear Equations and Functions
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, …}