Preview Warm Up California Standards Lesson Presentation

Warm Up 1. Solve 2x – 3y = 12 for y. 2. Evaluate the function f(x) = for –10, –5, 0, 5, and 10. f(–10) = –1 f(–5) = f(0) = 1 f(5) = 2 f(10) = 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

Vocabulary linear equation linear function

Many stretches on the German autobahn have a speed limit of 120 km/h 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 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.

Additional Example 1A: Graphing Linear Equations Graph y = 2x + 1. Tell whether it represents a function. Step 1 Choose three values of x and generate ordered pairs.  x y = 2x + 1 (x, y) y = 2(1) + 1 = 3 (1, 3) 1 y = 2(0) + 1 = 1 (0, 1) –1 y = 2(–1) + 1 = –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.

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 Subtract 15x from both sides. Since y is multiplied by 3 divide both sides by 3. y = –5x + 3

Additional Example 1B Continued Graph 15x + 3y = 9. Tell whether it represents a function. Step 2 Choose three values of x and generate ordered pairs  x y = –5x + 3 (x, y) 1 y = –5(1) + 3 = –2 (1, –2) y = –5(0) + 3 = 3 (0, 3) –1 y = –5(–1) + 3 = 8 (–1, 8) 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.

Additional 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.  Plot several points that have an x-coordinate of –2 and connect them with a straight line. There is a vertical line that intersects this graph more than once, so x = –2 does not represent a function.

Additional 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.

Graph y = 4x. Tell whether it represents a function. Check It Out! Example 1a Graph y = 4x. Tell whether it represents a function. Step 1 Choose three values of x and generate ordered pairs  x y = 4x (x, y) 1 y = 4(1) = 4 (1, 4) y = 4(0) = 0 (0, 0) –1 y = 4(–1) = –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.

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 Subtract x from both sides. y = –x + 7

Check 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  x (x, y) 1 –1 (1, 6) (0, 7) (–1, 8) y = –(1) + 7 = 6 y = –x + 7 y = –(0) + 7 = 7 y = –(–1) + 7 = 8 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.

Check It Out! Example 1c Graph . Tell whether it represents a function. Any ordered pair with an x-coordinate of will satisfy this equation.  Plot several points that have an x-coordinate of and connect them with a straight line. There is a vertical line that intersects this graph more than once, so x = does not describe a function.

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 2(3) + 5(2) 16 = ? 6 + 10 16 = ?  16 = 16 Since (3, 2) is a solution to 2x + 5y = 16, (3, 2) is on the graph.

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 2(2) + 5(2) 16 = ? 4 + 10 16 = ?  14  16 Since (2, 2) is not a solution to 2x + 5y = 16, (2, 2) is not on the graph.

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

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 5 – 3(1) 12 = ? 5 – 3 12 = ? 2  12  Since (5, 1) is not a solution to x – 3y = 12, (5, 1) is not on the graph.

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

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 1.5 – 3(–3.5) 12 = ? 1.5 + 10.5 12 = ?  12 = 12 Since (1.5, –3.5) is a solution to x – 3y = 12, (1.5, –3.5) is on the graph.

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 = 4 x + 0y = 4 The equation is in standard form. A = 1, B = 0, C = 4 The graph is a vertical line at x = 4.

Check It Out! Example 3a Write y = 5x – 9 in standard form and give the values of A, B, and C. Then describe the graph. y = 5x – 9 –5x –5x –5x + y = – 9 Subtract 5x from both sides. The equation is in standard form. A = –5, B = 1, C = –9 The graph is a line that is neither horizontal nor vertical.

Check It Out! Example 3b Write y = 12 in standard form and give the values of A, B, and C. Then describe the graph. y = 12 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.

Additional 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. 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}. 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)

Additional Example 4 Continued Graph the ordered pairs. Human Years vs. Dog Years  Any point on the line is a solution in this situation. The arrow shows that the trend continues. (6, 42)  (4, 28)  (2, 14)

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 Choose several values of x and make a table of ordered pairs. x f(x) = 3x + 10 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 The number of manicures must be a whole number, so the domain is {0, 1, 2, 3, …}. The range is {10, 13, 16, 19, …}.

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) no 4. (3, 5) 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, …}