Lesson 3.2 Graph Linear Equations Essential Question: How do you graph linear equations in the coordinate plane? 10-2-14 Warm-up: Common Core CC.9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. 1. Graph y = -x – 2 with the domain -2, -1, 0, 1, and 2 Rewrite the equation so y is a Function of x. 3x + 4y = 16 -6x – 2y = -12

Standardized Test Practice EXAMPLE 1 Standardized Test Practice Which ordered pair is a solution of 3x – y = 7? (3, 4) A (1, –4) B (5, –3) C (–1, –2) D SOLUTION Check whether each ordered pair is a solution of the equation. Test (3, 4): 3x – y = 7 Write original equation. 3(3) – 4 = ? 7 Substitute 3 for x and 4 for y. 5 = 7 Simplify.

Standardized Test Practice EXAMPLE 1 Standardized Test Practice Test (1, –4): 3x – y = 7 Write original equation. 3(1) – (–4) = ? 7 Substitute 1 for x and –4 for y. 7 = 7 Simplify. So, (3, 4) is not a solution, but (1, – 4) is a solution of 3x – y = 7. ANSWER The correct answer is B. A B D C

GUIDED PRACTICE for Example 1 Tell whether 4, – is a solution of x + 2y = 5. 1 2 not a solution ANSWER

EXAMPLE 2 Graph an equation Graph the equation –2x + y = –3. SOLUTION STEP 1 Solve the equation for y. –2x + y = –3 y = 2x –3

EXAMPLE 2 Graph an equation STEP 2 Make a table by choosing a few values for x and finding the values of y. x –2 –1 1 2 y –7 –5 –3 STEP 3 Plot the points. Notice that the points appear to lie on a line.

EXAMPLE 2 Graph an equation STEP 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

Standard Form of a Linear Equations The standard form of a linear equations is: Ax + By = C where A, B, and C are real numbers and A and B are not both zero.

EXAMPLE 3 Graph y = b and x = a Graph (a) y = 2 and (b) x = –1. SOLUTION For every value of x, the value of y is 2. The graph of the equation y = 2 is a horizontal line 2 units above the x-axis. a.

EXAMPLE 3 Graph y = b and x = a b. For every value of y, the value of x is –1. The graph of the equation x = –1 is a vertical line 1 unit to the left of the y-axis.

GUIDED PRACTICE for Examples 2 and 3 Graph the equation. 2. y + 3x = –2 ANSWER

GUIDED PRACTICE for Examples 2 and 3 Graph the equation. 3. y = 2.5 ANSWER

GUIDED PRACTICE for Examples 2 and 3 Graph the equation. 4. x = –4 ANSWER

Equations of Horizontal and Vertical Lines The graph of y = b is a horizontal line. The point passes through the point (0, b) The graph of x = a is vertical line that passes through the point (a,0)

Horizontal Line It is a function!

Vertical Line “y” values can be any numbers. However the line is not a function!

Linear Function What is a Linear Function?

Linear Function  

The graph of x = 5 is not a function. Why?

The graph of x = 5 is not a function. Why? Hint: Look at the graph!

EXAMPLE 4 Graph a linear function 1 2 Graph the function y = with domain x  0. – x + 4 Then identify the range of the function. SOLUTION STEP 1 Make a table. x 2 4 6 8 y 3 1

EXAMPLE 4 Graph a linear function STEP 2 Plot the points. STEP 3 Connect the points with a ray because the domain is restricted. STEP 4 Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4.

GUIDED PRACTICE for Example 4 5. Graph the function y = –3x + 1 with domain x  0. Then identify the range of the function. ANSWER y  1

EXAMPLE 5 Solve a multi-step problem RUNNING The distance d (in miles) that a runner travels is given by the function d = 6t where t is the time (in hours) spent running. The runner plans to go for a 1.5 hour run. Graph the function and identify its domain and range. SOLUTION STEP 1 Identify whether the problem specifies the domain or the range. You know the amount of time the runner plans to spend running. Because time is the independent variable, the domain is specified in this problem. The domain of the function is 0 ≤ t ≤ 1.5.

EXAMPLE 5 Solve a multi-step problem STEP 2 Graph the function. Make a table of values. Then plot and connect the points. t (hours) 0.5 1 1.5 d (miles) 3 6 9 STEP 3 Identify the unspecified domain or range. From the table or graph, you can see that the range of the function is 0 ≤ d ≤ 9.

EXAMPLE 6 Solve a related problem WHAT IF? Suppose the runner in Example 5 instead plans to run 12 miles. Graph the function and identify its domain and range. SOLUTION STEP 1 Identify whether the problem specifies the domain or the range. You are given the distance that the runner plans to travel. Because distance is the dependent variable, the range is specified in this problem. The range of the function is 0 ≤ d ≤ 12.

1 2 6 12 t (hours) d (miles) Solve a related problem EXAMPLE 6 STEP 2 Graph the function. To make a table, you can substitute d-values (be sure to include 0 and 12) into the function d = 6t and solve for t. t (hours) 1 2 d (miles) 6 12 STEP 3 Identify the unspecified domain or range. From the table or graph, you can see that the domain of the function is 0 ≤ t ≤ 2.

GUIDED PRACTICE for Examples 5 and 6 GAS COSTS 6. For gas that costs $2 per gallon, the equation C = 2g gives the cost C (in dollars) of pumping g gallons of gas. You plan to pump $10 worth of gas. Graph the function and identify its domain and range. ANSWER domain: 0 ≤ g ≤ 5, range: 0 ≤ C ≤ 10

Classwork/Homework 3.2 Practice B