Graphing a Function Rule

Goals Goal Rubric To graph equations that represent functions. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Continuous graph Discrete graph

Definition Some graphs are connected lines or curves called continuous graphs. Some graphs are only distinct points. They are called discrete graphs. Example discrete graphs: The graph on theme park attendance is an example of a discrete graph. It consists of distinct points because each year is distinct and people are counted in whole numbers only. The values between whole numbers are not included, since they have no meaning for the situation. Example continuous graphs: The graph of a car approaching a traffic light is an example of a continuous graph. It consists of continuous line and all the points on the line, because any point on the line has meaning. Car Approaching Traffic Light

Example: The graph is continuous. Henry begins to drain a water tank by opening a valve. Tell whether the graph is continuous or discrete. Water tank Water Level Time The graph is continuous.

Example: The graph is discrete. 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. Tell whether the graph is continuous or discrete. The graph is discrete.

Your Turn: A small bookstore sold between 5 and 8 books each day for 7 days. Tell whether the graph is continuous or discrete. The graph is discrete.

Your Turn: The graph is continuous. A truck driver enters a street, drives at a constant speed, stops at a light, and then continues. Tell whether the graph is continuous or discrete. Speed Time y x The graph is continuous.

Graphing Linear 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. 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.

Procedure: Graphing Functions Use the function to generate ordered pairs by choosing several values for x. Step 1 Step 2 Plot enough points to see a pattern for the graph. Step 3 Connect the points with a line or smooth curve.

Example: Graphing Linear Functions Graph y = 2x + 1. 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.

Example: Graphing Linear Functions Graph 15x + 3y = 9. Step 1 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 2 Plot the points and connect them with a straight line.

Your Turn: Graph y = 4x. x y = 4x (x, y) 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.

Your Turn: Graph y + x = 7. 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.

Graphing Real-World Functions In many real-world situations, the x and y values must be restricted. For example, quantities such as time, distance, and number of people can be represented using only nonnegative values. Sometimes x and y values 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.

Example: 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. Choose several values of x and make a table of ordered pairs. 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) The ages are continuous starting with 0.

Example: 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)

Your Turn: 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) = 3x + 10, where x is the number of manicures. Graph this function.

Your Turn: Solution The number of manicures must be a whole number. 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.

Your Turn: Solution Graph the ordered pairs. The individual points are solutions in this situation. The line shows that the trend continues.

Example: Graphing Nonlinear Functions Graph the function g(x) = |x| + 2. Step 1 Choose several values of x and generate ordered pairs. x g(x) = |x| + 2 (x, g(x)) –2 g(x) = |–2| + 2= 4 (–2, 4) –1 g(x) = |–1| + 2= 3 (–1, 3) g(x) = |0| + 2= 2 (0, 2) 1 g(x) = |1| + 2= 3 (1, 3) 2 g(x) = |2| + 2= 4 (2, 4) 3 g(x) = |3| + 2= 5 (3, 5)

Example: Continued Graph the function g(x) = |x| + 2. Step 2 Plot enough points to see a pattern.

Example: Continued Graph the function g(x) = |x| + 2. Step 3 The ordered pairs appear to form a v-shape. Draw lines through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on the “ends” of the “v”.

Your Turn: Graph the function y = |x – 1|. Step 1 Choose several values of x and generate ordered pairs. x y = |x – 1| (x, y) –2 y = |–2 – 1| = 3 (–2, 3) –1 y = |–1 – 1| = 2 (–1, 2) y = |0 – 1| = 1 (0, 1) 1 y = |1 – 1| = 0 (1, 0) 2 y = |2 – 1| = 1 (2, 1)

Continued Graph the function y = |x – 1|. Step 2 Plot enough points to see a pattern.

Continued Graph the function y = |x – 1|. Step 3 The ordered pairs appear to form a V-shape. Draw a line through the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the “V”..

Your Turn: Graph the function y = x2 + 1.

Solution: y = x2 + 1 x x2 + 1 y –2 –1 1 2 (–2)2 + 1 5 (–1)2 + 1 2 Plot the points and connect them with a smooth curve. x x2 + 1 y –2 –1 1 2 (–2)2 + 1 5 (–1)2 + 1 2 (0)2 + 1 1 (1)2 + 1 2 (2)2 + 1 5

Joke Time What do you call a man who lives in an envelope? Bill. What dog can't bark? A hot dog. What do you call a monkey on a mine field? A Ba-Boom.