EXAMPLE 2 COMMON CORE

Transcription

1 REFLECT. Why can it be helpful to solve a linear equation for y? Graphing a Linear Function Using the Slope and y-intercept You can graph the linear function f() = m + b using only the slope m and y-intercept b. First, locate the point (0, b) on the y-ais. Net, use the rise and run of the slope to locate another point on the line. Draw a line through the two points. Math On the Spot EXAMPLE COMMON CORE F.IF.7a Graph the function f() = - _ + 3 and determine its domain and range. STEP 1 The y-intercept is. Plot the point (0, ). f() STEP The slope is - _ 3. If you use - as the rise, then (0, ) the run is 3. (3, ) Use the slope to move from the y-intercept to a second point. Begin - - O by moving down - units, because the rise is negative. Then move - right 3 units because the run is positive. Plot the second point, (3, ). STEP 3 STEP Draw a line through the two points. The domain is the set of real numbers. The range is the set of real numbers. REFLECT 3. Multiple Representations How does the graph of the linear function show that the domain is the set of real numbers and the range is the set of real numbers? Lesson. 11

2 YOUR TURN Personal Math Trainer Online Practice and Help Graph each function.. f() = - 1_ f() = - 1 f() f() - - O - - O Modeling with Slope-Intercept Form Many real-world situations can be modeled by linear equations in slope-intercept form. Math On the Spot EXAMPLE 3 COMMON CORE F.IF., F.LE.1b A pitcher with a maimum capacity of cups contains 1 cup of apple juice concentrate. A faucet is turned on, filling the pitcher at a rate of 0.5 cup per second. The amount of liquid in the pitcher, A(t), (in cups), is a function of the time t (in seconds) that the water is running. Graph the function A(t), write the rule for the function, and state its domain and range. STEP 1 The y-intercept is 1 because there is 1 cup in the pitcher at time 0. Plot the point that corresponds to the y-intercept, (0, 1). STEP The slope is the rate of change: 0.5 cup per second, or 1 cup in seconds. So the rise is 1 and the run is. STEP 3 STEP Use the rise and run to move from the first point to a second point on the line by moving up 1 unit and right units. Plot the second point, (, ). Connect the points and etend the line segment to the maimum value of the function, where A(t) = cups. Amount of liquid (cups) A(t) O 1 1 Time (seconds) t 1 Unit A

3 STEP 5 Use 1_ 1_ and b = 1 to write the rule for the function: A(t) = t + 1. STEP The domain is the set of all real numbers t such that 0 t 1. The range is the set of all real numbers A(t) such that 1 A(t). REFLECT. Critical Thinking Why are the domain and range restricted in Eample 3, rather than each being the set of all real numbers? YOUR TURN 7. A pump is set to dispense chlorine from a full 5-gallon container into a swimming pool to sanitize the water. The pump will dispense the chlorine at a rate of 0.5 gallon per minute and will shut off when the container is empty. The amount of chlorine in the container, A(t), (in gallons), is a function of the time t (in minutes) that the pump is running. Graph the function A(t), write the rule for the function, and state its domain and range. A(t) A(t) = Amount of chlorine (gal) O Time (min) 10 1 t Domain: Range: Personal Math Trainer Online Practice and Help Lesson. 13

4 Guided Practice Find the slope of the line described by each equation. (Eample 1) y = 10. 3y = y =. + y = 1 5. Graph the function f() = and determine its domain and range. (Eample ) STEP 1 The y-intercept is. f() Plot the point (, ). STEP The slope is. Use as the rise; then the run is. Use the slope to move from the y-intercept to a second point. Begin by moving - - O - unit(s). Then move unit(s). - Plot the second point, (, ). STEP 3 Draw a line through the two points. STEP The domain is the set of numbers. The range is the set of numbers.? ESSENTIAL QUESTION CHECK-IN. How is the rate of change in a real-world linear relationship related to the slope-intercept form of the equation that represents the relationship? 1 Unit A

5 Name Class Date. Independent Practice COMMON CORE F.IF., F.IF., F.IF.7, F.IF.7a, F.LE.1b, A.CED. Personal Math Trainer Online Practice and Help Find the slope of the line described by each equation y = 0. 3y = y = When graphing a linear function in slope-intercept form, why do you have to plot the y-intercept first? Why can t you use the slope first? Graph each linear function. 11. f() = 1_ f() = f() = -1 f() f() f() - - O O O A company rents moving vans for a charge of $30 plus $0.50 per mile. The company only allows its vans to be used for in-town moves, with total mileage limited to 100 miles. The total rental cost, C(m), (in dollars) is a function of the distance m (in miles) that the van is driven. State a rule for the function, graph the function, and state its domain and range. C(m) = Domain: Range: Cost ($) O C(m) Distance (mi) m Lesson. 15

6 FOCUS ON HIGHER ORDER THINKING Work Area 15. Draw Conclusions The standard form of a linear equation is A + By = C. Rewrite this equation in slope-intercept form. What is the slope? What is the y-intercept? 1. What If? What if the person filling the pitcher in Eample 3 gets distracted by a phone call and does not get to turn the faucet off as soon as the pitcher is full? How does this affect the domain and range of the function? How does it affect the graph? 17. Find the Error Alyssa correctly determines that the graph of a linear equation intersects the -ais at (, 0) and intersects the y-ais at (0, ). She calculates the slope and then writes the slope-intercept equation for the line as y = - 1_ 3 +. What error did Alyssa make? What is the correct slope-intercept equation for this line? 1. Justify Reasoning Is it possible to write the equation of every line in slope-intercept form? Eplain your reasoning. 1 Unit A